{"id":90191,"date":"2022-03-30T15:20:58","date_gmt":"2022-03-30T09:50:58","guid":{"rendered":"https:\/\/bigideasmathanswers.com\/?p=90191"},"modified":"2022-04-07T14:31:25","modified_gmt":"2022-04-07T09:01:25","slug":"envision-math-common-core-grade-8-answer-key-topic-4","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/envision-math-common-core-grade-8-answer-key-topic-4\/","title":{"rendered":"enVision Math Common Core Grade 8 Answer Key Topic 4 Investigate Bivariate Data"},"content":{"rendered":"

Practice with the help of enVision Math Common Core Grade 8 Answer Key<\/a> Topic 4 Investigate Bivariate Data<\/strong> regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 8th Grade Answers Key Topic 4 Investigate Bivariate Data<\/h2>\n

3-ACT MATH<\/strong><\/p>\n

Reach Out<\/strong><\/p>\n

Reach for the skies! Who in your class can reach the highest? That height depends on how tall each person is and the lengths of their arms.
\nNow stick your arms out to your sides. Sometimes this horizontal distance is called your wingspan. The wandering albatross can have a wingspan of up to 12 feet. How does your wingspan compare? Think about this during the 3-Act Mathematical Modeling lesson.
\n\"Envision<\/p>\n

enVision STEM Project<\/strong><\/p>\n

Did You Know?<\/strong>
\nA fishery biologist collects data on fish, such as the size and health of the fish population in a particular body of water.
\n\"Envision
\nLargemouth bass and smallmouth bass are the most popular game fish in North America.
\n\"Envision
\nBiologists often use tagging studies to estimate fish population, as well as to estimate catch and harvest rates.
\n\"Envision
\nThe average lifespan of bass is about 16 years, but some have lived more than 20 years.<\/p>\n

Research suggests that bass can see red better than any other color on the spectrum.
\n\"Envision<\/p>\n

Your Task: How Many Fish?<\/strong><\/p>\n

Suppose a fishery biologist takes 500 basses from a lake, tags them, and then releases them back into the water. Several days later, the biologist nets a sample of 200 basses, of which 30 are tagged. How many basses are in the lake? You and your classmates will explore how the biologist can use sampling to describe patterns and to make generalizations about the entire population.
\n\"Envision
\nAnswer:
\nIt is given that
\nA fishery biologist takes 500 basses from a lake, tags them, and then releases them back into the water. Several days later, the biologist nets a sample of 200 basses, of which 30 are tagged.
\nSo,
\nThe total number of basses = 500 + 200
\n= 700
\nThe number of basses tagged = 30
\nSo,
\nAfter netting, the number of basses = 700 – 30
\n= 670
\nHence, from the above,
\nWe can conclude that after the biologist nets 200 basses,
\nThe total number of basses present are: 670<\/p>\n

Topic 4 GET READY!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong><\/p>\n

Choose the best term from the box to complete each definition.<\/p>\n

Question 1.
\n____ is the change in y divided by the change in x.
\nAnswer:
\nWe know that,
\n“Slope” is the change in y divided by the change in x
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Slope<\/p>\n

Question 2.
\nA relationship where for every x units of one quantity there are y units of another quantity is a ____
\nAnswer:
\nWe know that,
\nA relationship where for every x units of one quantity, there are y units of another quantity is a “Ratio”
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Ratio<\/p>\n

Question 3.
\nThe ____ is the horizontal line in a coordinate plane.
\nAnswer:
\nWe know that,
\nThe “X-axis” is the horizontal line in a coordinate plane
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: X-axis<\/p>\n

Question 4.
\nThe ___ is the vertical line in a coordinate plane.
\nAnswer:
\nWe know that,
\nThe “Y-axis” is the vertical line in a coordinate plane
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Y-axis<\/p>\n

Graphing Points<\/strong>
\nGraph and label each point on the coordinate plane.<\/p>\n

\"Envision<\/p>\n

Question 5.
\n(-2, 4)
\nAnswer:
\n\"\"<\/p>\n

Question 6.
\n(0, 3)
\nAnswer:
\n\"\"<\/p>\n

Question 7.
\n(3, -1)
\nAnswer:
\n\"\"<\/p>\n

Question 8.
\n(-4, -3)
\nAnswer:
\n\"\"<\/p>\n

Finding Slope<\/strong><\/p>\n

Find the slope between each pair of points.<\/p>\n

Question 9.
\n(4, 6) and (-2, 8)
\nAnswer:
\nThe given points are: (4, 6), (-2, 8)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{8 – 6}{-2 – 4}\\)
\n= \\(\\frac{2}{-6}\\)
\n= –\\(\\frac{1}{3}\\)
\nHence, from the above,
\nWe can conclude that the slope between the given points is: –\\(\\frac{1}{3}\\)<\/p>\n

Question 10.
\n(-1, 3) and (5,9)
\nAnswer:
\nThe given points are: (-1, 3), (5, 9)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{9 – 3}{5 + 1}\\)
\n= \\(\\frac{6}{6}\\)
\n= 1
\nHence, from the above,
\nWe can conclude that the slope between the given points is: 1<\/p>\n

Question 11.
\n(5, -1) and (-3, -7)
\nAnswer:
\nThe given points are: (5, -1), (-3, -7)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{-7 + 1}{-3 – 5}\\)
\n= \\(\\frac{-6}{-8}\\)
\n= \\(\\frac{3}{4}\\)
\nHence, from the above,
\nWe can conclude that the slope between the given points is: \\(\\frac{3}{4}\\)<\/p>\n

Writing Fractions as Percents<\/p>\n

Question 12.
\nExplain how to write \\(\\frac{36}{60}\\) as a percent.
\nAnswer:
\nThe given fraction is: \\(\\frac{36}{60}\\)
\nWe know that,
\n1 = 100%
\nSo,
\n\\(\\frac{36}{60}\\)
\n= \\(\\frac{36}{60}\\) \u00d7 100%
\n= \\(\\frac{36 \u00d7 100%}{60}\\)
\n= 60%
\nHence, from the above,
\nWe can conclude that the value of \\(\\frac{36}{60}\\) as a percent is: 60%<\/p>\n

Language Development<\/strong><\/p>\n

Complete the graphic organizer. Write the definitions of the terms in your own words. Use words or a sketch to show an example.<\/p>\n

\"Envision
\n\"Envision
\n\"Envision
\nAnswer:
\n\"\"<\/p>\n

Topic 4 PICK A PROJECT<\/h3>\n

PROJECT 4A<\/strong><\/p>\n

What carnival games do you have a good chance of winning, and why?
\nPROJECT: BUILD A CARNIVAL GAME
\n\"Envision<\/p>\n

PROJECT 4B<\/strong>
\nIf you had a superpower, what would it be?
\nPROJECT: SUMMARIZE SUPERHERO DATA
\n\"Envision<\/p>\n

PROJECT 4C<\/strong>
\nWhat makes a song’s lyrics catchy?
\nPROJECT: WRITE A SONG
\n\"Envision<\/p>\n

PROJECT 4D<\/strong>
\nHow does your dream job use math?
\nPROJECT: RESEARCH A CAREER
\n\"Envision<\/p>\n

Lesson 4.1 Construct and Interpret Scatter Plots<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Luciana is starting a two-week social media campaign to attract new subscribers to Blaston, a music website for teens. She has the following data from her last campaign to help plan her strategy.<\/p>\n

\"Envision<\/p>\n

Look for Relationships<\/strong>
\nHow are the number of media posts and the number of subscribers related?
\nAnswer:
\nThe given data is:
\n\"Envision
\nIf we observe the total data,
\nWe can find that there is no particular pattern for the given data
\nBut, if we observe the given data in parts, then
\nFrom 1 – 5 posts per day,
\nSocial media posts per day \u221d New subscribers
\nFrom 8 – 10 posts per day,
\nSocial media posts per day \u221d \\(\\frac{1}{New subscribers}\\)
\nFrom 6 – 7 posts per day,
\nThere is no pattern between the social media posts per day and the number of subscribers<\/p>\n

Based on this data, what should be Luciana’s strategy for the new campaign?
\nAnswer:
\nBased on the pattern of the given data (As mentioned above),
\nLuciana’s strategy for the new campaign must be:
\nThe social media posts per day and the new subscribers must be in an increasing trend only<\/p>\n

Focus on math practices<\/strong>
\nUse Structure What patterns do you see in the data from Luciana’s last social media campaign?
\nAnswer:
\nIf we observe the total data,
\nWe can find that there is no particular pattern for the given data
\nBut, if we observe the given data in parts, then
\nFrom 1 – 5 posts per day,
\nSocial media posts per day \u221d New subscribers
\nFrom 8 – 10 posts per day,
\nSocial media posts per day \u221d \\(\\frac{1}{New subscribers}\\)
\nFrom 6 – 7 posts per day,
\nThere is no pattern between the social media posts per day and the number of subscribers<\/p>\n

Essential Question<\/strong>
\nHow does a scatter plot show the relationship between paired data?
\nAnswer:
\nThe\u00a0scatter\u00a0diagram graphs\u00a0pairs of\u00a0numerical\u00a0data, with one variable on each axis,\u00a0to\u00a0look for a\u00a0relationship between\u00a0them. If the variables\u00a0are\u00a0correlated, the points will fall along a line or curve. The better the\u00a0correlation, the tighter the points will hug the line.<\/p>\n

Try It!<\/strong>
\nLuciana collects data about the number of entries and the ages of the subscribers who enter the concert giveaway.<\/p>\n

\"Envision<\/p>\n

The point that represents the data in the fourth column has coordinates \"Envision
\nAnswer:
\nThe given data is:
\n\"Envision
\nNow,
\nWe know that,
\nThe x-axis always represents the independent variables
\nThe y-axis always represents the dependent variables
\nSo,
\nFrom the given data,
\nThe independent variable (x) is: Age
\nThe dependent variable (y) is: Number of entries
\nWe know that,
\nThe ordered pair must be in the form of (x, y)
\nHence, from the above,
\nWe can conclude that the point that represents the data in the fourth column has coordinates (13, 9)<\/p>\n

Convince Me!<\/strong>
\nExplain how Luciana would choose scales for the x-axis and y-axis.
\nAnswer:
\nIn a graph,
\nThe scales for the x-axis and the y-axis is nothing but the rate of change between the values of x and y
\nEx:
\nThe given data is:
\n\"Envision
\nFrom the above data,
\nThe scale for the x-axis is:
\nRate of change between the values of x = 11 – 10 (or) 12 – 11 and so on
\n= 1
\nThe scale for the y-axis is:
\nRate of change between the values of y = 9 – 8 (or) 10 – 9
\n= 1
\nHence, from the above,
\nWe can conclude that
\nThe scale for the x-axis is:
\n1 unit = 1 year
\nThe scale for the y-axis is:
\n1 unit = 1 Entry<\/p>\n

Try It!<\/strong>
\nAvery also tracks the number of minutes a player plays and the number of points the player scored. Describe the association between the two data sets. Tell what the association suggests.
\n\"Envision
\nAnswer:
\nIt is given that
\nAvery also tracks the number of minutes a player plays and the number of points the player scored.
\nSo,
\nThe given scatter plot is:
\n\"Envision
\nNow,
\nFrom the given scatter plot,
\nWe can observe that the overall pattern is in an increasing trend
\nBut,
\nWhen we observe the number of points in the perspective of minutes,
\nThe pattern is in clusters
\nNow,
\nAt 4 minutes,
\nThe number of points scored is: 0
\nAt 6 and 8 minutes,
\nThe number of points scored is: 2
\nBetween 8 and 14 minutes,
\nThe number of points scored is: 4
\nBetween 10 and 16 minutes,
\nThe number of points scored is: 6<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

A scatter plot shows the relationship, or association, between two sets of data.
\n\"Envision
\nThe y-values increase as the x-values increase.
\n\"Envision
\nThe y-values decrease as the x-values increase.
\n\"Envision
\nThere is no consistent pattern between the y-values and the x-values.<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong>
\nHow does a scatter plot show the relationship between paired data?
\nAnswer:
\nThe\u00a0scatter\u00a0diagram graphs\u00a0pairs of\u00a0numerical\u00a0data, with one variable on each axis,\u00a0to\u00a0look for a\u00a0relationship between\u00a0them. If the variables\u00a0are\u00a0correlated, the points will fall along a line or curve. The better the\u00a0correlation, the tighter the points will hug the line.<\/p>\n

Question 2.
\nModel with Math<\/strong>
\nMarcy always sleeps fewer than 9 hours each night and has never scored more than 27 points in a basketball game. A scatter plot suggests that the more sleep she gets, the more she scores. What scales for the axes might be best for constructing the scatter plot?
\nAnswer:
\nIt is given that
\nMarcy always sleeps fewer than 9 hours each night and has never scored more than 27 points in a basketball game. A scatter plot suggests that the more sleep she gets, the more she scores.
\nSo,
\nThe scales for the axes that might be best is:
\nFor the x-axis:
\nIt is given that Macy always sleeps fewer than 9 hours
\nSo,
\nThe scale might be: 1 unit starting from 9 to 24
\nFor the y-axis:
\nIt is given that Macy never scored more than 27 points in a basketball game
\nSo,
\nThe scale might be: 1 unit starting from 27 to the corresponding last value of x
\nHence, from the above,
\nWe can conclude that
\nFor the x-axis,
\nThe scale is:
\n1 unit = 1 hour
\nFor the y-axis,
\nThe scale is:
\n1 unit = 1 point<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Kyle says that every scatter plot will have a cluster, gap, and outlier. Is he correct? Explain.
\nAnswer:
\nWe know that,
\nA scatter plot might have a cluster, a gap, and an outlier or the association of any two but not all three are present
\nHence, from the above,
\nWe can conclude that Kyle is not correct<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nPhoebe constructs a scatter plot to show the data. What scales could she use for the x- and y-axes?
\n\"Envision
\nAnswer:
\nIt is given that
\nPhoebe constructs a scatter plot to show the data.
\nNote:
\nThe scatter plot is drawn only for the relations that are functions and we know that the rate of change is constant for a function
\nNow,
\nThe given data is:
\n\"Envision
\nFrom the given data,
\nThe scale she could use for the x-axis will be 1 unit
\nThe scale she might use for the y-axis will be: 1 unit
\nHence, from the above,
\nWe can conclude that
\nThe scale used for the x-axis is:
\n1 unit = 1 inch (Shoe size)
\nThe scale used for the y-axis is:
\n1 unit = 1 inch (Height)<\/p>\n

Question 5.
\nGermaine constructs a scatter plot to show how many people visit different theme parks in a month. Why might clusters and outliers be present?
\n\"Envision
\nAnswer:
\nIt is given that
\nGermaine constructs a scatter plot to show how many people visit different theme parks in a month
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that the graph is non-linear
\nWe know that,
\nA\u00a0cluster\u00a0is a group of objects, numbers, data points (information), or even people that are located close together
\nAn\u00a0outlier\u00a0is a value in a data set that is very different from the other values. That is,\u00a0outliers\u00a0are values unusually far from the middle
\nSo,
\nFrom the given scatter plot,
\nWe can observe that there are 2 points that are far from the pattern and those points are called “Outliers”
\nSince the scatter plot is non-linear, the points will be grouped together and the group of points is called “Clusters”<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 6.
\nLeveled Practice<\/strong> The table shows the racing times in minutes for the first two laps in a race. Complete the scatter plot.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe table shows the racing times in minutes for the first two laps in a race
\nNow,
\nIn the graph,
\nThe x-axis represents: The racing times of Lap 1
\nThe y-axis represents: The racing times of Lap 2
\nHence,
\nThe completed scatter plot with the x-axis and y-axis is:
\n\"\"<\/p>\n

Question 7.
\nThe scatter plot represents the prices and number of books sold in a bookstore.
\na. Identify the cluster in the scatter plot and explain what it means.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe scatter plot represents the prices and number of books sold in a bookstore.
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nNow,
\nWe know that,
\nA\u00a0cluster\u00a0is a group of objects, numbers, data points (information), or even people that are located close together
\nSo,
\nFrom the above scatter plot,
\nWe can observe that the clusters are present between the intervals of 10 books sold and 20 books sold<\/p>\n

b. Generalize<\/strong> How does the scatter plot show the relationship between the data points? Explain.
\nAnswer:
\nThe\u00a0<\/span>scatter\u00a0diagram graphs\u00a0<\/span>pairs of\u00a0numerical\u00a0<\/span>data, with one variable on each axis,\u00a0<\/span>to\u00a0look for a\u00a0<\/span>relationship between\u00a0them. If the variables\u00a0<\/span>are correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line.<\/span><\/p>\n

Question 8.
\nThe table shows the monthly attendance in thousands at museums in one country over a 12-month period.
\n\"Envision
\na. Complete the scatter plot to represent the data.
\nAnswer:
\nIt is given that
\nThe table shows the monthly attendance in thousands at museums in one country over a 12-month period.
\nNow,
\nThe given data is:
\n\"Envision
\nHence,
\nThe representation of the scatter plot for the given data is:
\n\"\"<\/p>\n

b. Identify any outliers in the scatter plot.
\nAnswer:
\nWe know that,
\nA value that “lies outside” (is much smaller or larger than) most of the other values in a set of data is called an “Outlier”
\nHence, from the above,
\nWe can conclude that
\nThe outliers in the scatterplot are at (12, 3), (6, 36)<\/p>\n

c. What situation might have caused an outlier?
\n\"Envision
\nAnswer:
\nFrom part (b),
\nThe outliers in the scatterplot are at (12, 3), (6, 36)
\nSo,
\nFrom the given points,
\nWe can conclude that the number of people is very low at that particular month to cause the situation of outliers<\/p>\n

Question 9.
\nHigher-Order Thinking<\/strong> The table shows the number of painters and sculptors enrolled in seven art schools. Jashar makes an incorrect scatter plot to represent the data.
\n\"Envision
\na. What error did Jashar likely make?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe table shows the number of painters and sculptors enrolled in seven art schools.
\nNow,
\nThe given data is:
\n\"Envision
\nSo,
\nFrom the given data,
\nWe can observe that
\nThe variable that will be on the x-axis (Independent variable) is: Number of painters
\nThe variable that will be on the y-axis (Dependent variable) is: Number of sculptors
\nBut,
\nFrom the scatter plot drawn by Jashar,
\nHe interchanged the variables of the axes i.e., he took the independent variable at the y-axis and the dependent variable at the x-axis
\nHence, from the above,
\nWe can conclude that the error made by Jashar is the interchange of the variables of the axes<\/p>\n

b. Explain the relationship between the number of painters and sculptors enrolled in the art schools.
\n\"Envision
\nAnswer:
\nFrom the given data,
\nWe can observe that for the increase in painters, the number of sculptors also increases
\nHence, from the above,
\nWe can conclude that the relationship between the number of painters and sculptors enrolled in the art schools is:
\nNumber of painters \u221d Number of sculptors<\/p>\n

c. Reasoning<\/strong> Jashar’s scatter plot shows two possible outliers. Identify them and explain why they are outliers.
\nAnswer:
\nWe know that,
\nA value that “lies outside” (is much smaller or larger than) most of the other values in a set of data is called an “Outlier”
\nNow,
\nThe scatter plot for the given data is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the two possible outliers identified by Jashar are: (11, 6), and (20, 45)<\/p>\n

Assessment Practice<\/strong><\/p>\n

Use the scatter plot to answer 10 and 11.<\/p>\n

Question 10.
\nTen athletes in the Florida Running Club ran two races of the same length. The scatter plot shows their times. Select all statements that are true.
\n\"Envision
\n\"Envision Nine of the times for the first race were at least 16 seconds.
\n\"Envision Eight of the times for the second race were less than 17 seconds.
\n\"Envision There were seven athletes who were faster in the second race than in the first.
\n\"Envision There were three athletes who had the same time in both races.
\n\"Envision There were three athletes whose times in the two races differed by exactly 1 second.
\nAnswer:
\nIt is given that
\nTen athletes in the Florida Running Club ran two races of the same length. The scatter plot shows their times.
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nHence,
\nThe correct statements about the given scatter plot is:\"\"<\/p>\n

Question 11.
\nWhat was the greatest difference for a single runner in finishing times in the races?
\nA. 3 seconds
\nB. 4 seconds
\nC. 5 seconds
\nD. 7 seconds
\nAnswer:
\nFrom the given scatter plot,
\nWe can observe that
\nThe lowest time a runner takes for completing a race is approximately 14 seconds
\nThe highest time a runner takes for completing a race is approximately 17 seconds
\nSo,
\nThe greatest difference for a single runner in finishing times in the races is: 3 seconds
\nHence, from the above,
\nWe can conclude that option A matches the given situation<\/p>\n

Lesson 4.2 Analyze Linear Associations<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Angus has a big test coming up. Should he stay up and study or go to bed early the night before the test? Defend your recommendation.
\n\"Envision
\nAnswer:
\nIt is given that
\nAngus has a big test coming up
\nNow,
\nThe relationship between the sleeping time and the percentage of marks is also given
\nNow,
\nFrom the given data,
\nWe can observe that
\nIf he went to bed early i.e., at 9:00, then he got 93%
\nIf he studied until 11:00, then he got only 92%
\nHence, from the above,
\nWe can conclude that Angus has to go to bed early before the big test<\/p>\n

Generalize
\n<\/strong>Can you make a general statement about which option leads to a better result?
\nAnswer:
\nGenerally, going to bed early will lead to better results before a test<\/p>\n

Focus on math practices<\/strong>
\nConstruct Arguments<\/strong> What other factors should Angus also take into consideration to make a decision? Defend your response.
\nAnswer:
\nThe other factors that Angus should also take into consideration when making a decision are:
\nA) Nature of the exam
\nB) Coverage of the topics for the particular exam
\nC) Number of revisions<\/p>\n

? Essential Question<\/strong>
\nHow can you describe the association of two data sets?
\nAnswer:
\nAssociation (or relationship) between two variables will be described as strong, weak, or none; and the direction of the association may be positive, negative, or none<\/p>\n

Try It!<\/strong>
\nGeorgia and her classmates also measured their foot length. Use a pencil to find the trend line. Sketch the trend line for the scatter plot.
\n\"Envision
\nAnswer:
\nIt is given that
\nGeorgia and her classmates also measured their foot length
\nHence,
\nThe representation of a trend line for the given scatter plot is:
\n\"\"<\/p>\n

Try It!<\/strong>
\nFor each scatter plot, identify the association between the data. If there is no association, state so.
\na.
\n\"Envision<\/p>\n

Answer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the above scatter plot,
\nWe can observe that the points are all scattered
\nHence, from the above,
\nWe can conclude that the given scatter plot has a weaker association<\/p>\n

b.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nThe points are all in a non-linear shape
\nHence, from the above,
\nWe can conclude that the given scatter plot has a non-linear association
\nc.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the scatter plot,
\nWe can observe that the points are all scattered
\nHence, from the above,
\nWe can conclude that the given scatter plot has a weaker association<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

Scatter plots can show a linear association, a nonlinear association, or no association. For scatter plots that suggest a linear association, you can draw a trend line to show the association. You can assess the strength of the association by looking at the distances of plotted points from the trend line.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you describe the relationship between the two sets of data?
\nAnswer:
\nAssociation (or relationship) between two variables will be described as strong, weak, or none; and the direction of the association may be positive, negative, or none<\/p>\n

Question 2.
\nLook for Relationships<\/strong> How does a trend line describe the strength of the association?
\nAnswer:
\nThe straight\u00a0<\/span>line\u00a0is a\u00a0<\/span>trend line, designed to come as close as possible to all the data points. The\u00a0<\/span>trend line\u00a0has a positive slope, which shows a positive\u00a0<\/span>relationship\u00a0between X and Y. The points in the\u00a0<\/span>graph\u00a0are tightly clustered about the\u00a0<\/span>trend line\u00a0due to the strength of the\u00a0<\/span>relationship between X and Y.<\/span><\/p>\n

Question 3.
\nConstruct Arguments<\/strong> How does the scatter plot of a nonlinear association differ from that of a linear association?
\nAnswer:
\nScatterplots with a\u00a0linear\u00a0pattern have points that seem to generally fall along a line while\u00a0nonlinear\u00a0patterns seem to follow along some curve. Whatever the pattern is, we use this to describe the\u00a0association\u00a0between the variables.<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nDescribe the association between the two sets of data in the scatter plot.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that all the points are tightly hugged by a trend line
\nHence, from the above,
\nWe can conclude that the given scatter plot has a stronger association<\/p>\n

Question 5.
\nDescribe the association between the two sets of data in the scatter plot.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that the points are all in a non-linear shape and are closely connected
\nHence, from the above,
\nWe can conclude that the given scatter plot has a non-linear association<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/p>\n

Question 6.
\nThe scatter plot shows the average heights of children ages 2-12 in a certain country. Which line is the best model of the data?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe scatter plot shows the average heights of children ages 2-12 in a certain country.
\nNow,
\nThe best line in the given scatter plot is that line that tightly hugs the maximum points in a scatter plot
\nHence, from the above,
\nWe can conclude that line m is the best model of the given data<\/p>\n

Question 7.
\nDoes the scatter plot shows a positive, a negative, or no association?
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that as the value of x increases, the value of y also increases
\nHence, from the above,
\nWe can conclude that the given scatter plot has a positive association<\/p>\n

Question 8.
\nDetermine whether the scatter plot of the data for the following situation would have a positive or negative linear association.
\ntime working and amount of money earned
\nAnswer:
\nThe given situation is:
\nTime working and amount of money earned
\nWe know that,
\nThe total amount of work done = Number of days \u00d7 The amount earned for the work done
\nLet us suppose the number of days is constant
\nSo,
\nThe total amount of work done \u221d The amount earned for the work done
\nSo,
\nThe more time a person works, the more money that person will earn
\nHence, from the above,
\nWe can conclude that the scatter plot of the given data has a positive linear association<\/p>\n

Question 9.
\nDescribe the relationship between the data in the scatter plot.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that the data in the scatter plot has a decreasing trend with the strong association of data with each other
\nHence, from the above
\nWe can conclude that the given scatter plot has a negative linear association<\/p>\n

Question 10.
\nDescribe the relationship between the data in the scatter plot.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that all the points are in a cyclic fashion
\nHence, from the above,
\nWe can conclude that the given scatter plot has a non-linear association<\/p>\n

Question 11.
\nHigher-Order Thinking<\/strong> Describe a real situation that would fit the relationship described.
\na. A strong, positive association
\nAnswer:
\nThe real-life examples for a strong, positive association are:
\nA) The more time you spend running on a treadmill, the more calories you will burn.
\nB) Taller people have larger shoe sizes and shorter people have smaller shoe sizes.
\nC) The longer your hair grows, the more shampoo you will need.
\nD) The less time I spend marketing my business, the fewer new customers I will have.
\nE) The more hours you spend in direct sunlight, the more severe your sunburn.<\/p>\n

b. A strong, negative association
\nAnswer:
\nThe real-life examples for a strong, negative association are:
\nA) A student who has many absences has a decrease in grades.
\nB) As the weather gets colder, air conditioning costs decrease.
\nC) If a train increases speed, the length of time to get to the final point decreases.
\nD) If a chicken increases in age, the number of eggs it produces decreases.
\nE) If the sun shines more, a house with solar panels requires less use of other electricity.<\/p>\n

Question 12.
\nA sociologist is studying how sleep affects the amount of money a person spends. The scatter plot shows the results of the study. What type of association does it show between the amount of sleep and money spent?
\n\"Envision
\nAnswer:
\nIt is given that
\nA sociologist is studying how sleep affects the amount of money a person spends. The scatter plot shows the results of the study
\nNow,
\nFrom the given scatter plot,
\nWe can observe that the data that is related to the amount of sleep and the amount of money spent is in a cyclic fashion
\nHence, from the above,
\nWe can conclude that the association does it show between the amount of sleep and money spent is: Non-linear association<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 13.
\nWhich paired data would likely show a positive association? Select all that apply.
\n\"Envision Population and the number of schools
\n\"Envision Hair length and shoe size
\n\"Envision Number of people who carpool to work and money spent on gas
\n\"Envision Hours worked and amount of money earned
\n\"Envision Time spent driving and amount of gas in the car
\nAnswer:
\nWe know that,
\nA positive association is an association that as the value of x increases, the value of y also increases
\nHence,
\nThe paired data that would likely show a positive association is\"\"<\/p>\n

Question 14.
\nWhich paired data would likely show a negative association? Select all that apply.
\n\"Envision Population and the number of schools
\n\"Envision Hair length and shoe size
\n\"Envision Number of people who carpool to work and money spent on gas
\n\"Envision Hours worked and amount of money earned
\n\"Envision Time spent driving and amount of gas in the car
\nAnswer:
\nWe know that,
\nA negative association is an association that as the value of x increases, the value of y also decreases
\nHence,
\nThe paired data that would likely show a negative association is:
\n\"\"<\/p>\n

Lesson 4.3 Use Linear Models to Make Predictions<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Bao has a new tracking device that he wears when he exercises. It sends data to his computer. How can Bao determine how long he should exercise each day if he wants to burn 5,000 Calories per week?
\n\"Envision
\nAnswer:
\nIt is given that
\nBao has a new tracking device that he wears when he exercises. It sends data to his computer
\nNow,
\nIt is also given that Bao wants to burn 5,000 calories per week
\nSo,
\nThe number of calories Bao wants to burn per day = \\(\\frac{5,000}{7}\\)
\n= 714.2 calories
\n= 714 calories
\n\u2245 720 calories
\nNow,
\nFrom the given scatter plot,
\nWe can observe that
\nFor approximately 720 calories to burn, Bao has to exercise 80 – 90 minutes each day
\nHence, from the above,
\nWe can conclude that Bao should exercise 80 – 90 minutes each day if he wants to burn 5,000 Calories per week<\/p>\n

Focus on math practices<\/strong><\/p>\n

Reasoning<\/strong> Suppose another set of data were plotted with a trend line passing through (25, 100) and (80, 550). Would this indicate that more or fewer calories were burned per minute? Explain.
\nAnswer:<\/p>\n

? Essential Question<\/strong>
\nHow do linear models help you to make a prediction?
\nAnswer:
\nWhile\u00a0linear models\u00a0do not always accurately represent\u00a0data, and this occurs when actual\u00a0data\u00a0does not clearly show a relationship between its two variables,\u00a0linear models\u00a0are\u00a0helpful\u00a0in determining the future points of\u00a0data, the expected points of\u00a0data, and the highest possible accuracy of\u00a0data.<\/p>\n

Try It!<\/strong><\/p>\n

Assuming the trend shown in the graph continues, use the equation of the trend line to predict average fuel consumption in miles per gallon in 2025.
\nThe equation of the trend line is y = \"Envisionx + \"Envision. In 2025, the average fuel consumption is predicted to be about mpg.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the scatter plot,
\nWe can observe that
\nThe initial value (y-intercept) is: 15
\nNow,
\nWe know that,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the initial value (or) y-intercept
\nNow,
\nTo find the slope,
\nThe points from the given scatter plot is: (15, 21), (30, 24)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{24 – 21}{30 – 15}\\)
\n= \\(\\frac{1}{5}\\)
\nSo,
\nThe equation of the line is:
\ny = 0.20x + 15
\nNow,
\nThe average fuel consumption in 2025 is:
\ny = 0.20 (2025 – 1980) + 15
\ny = 0.20 (45) + 15
\ny = 9 + 15
\ny = 24 mpg
\nHence, from the above,
\nWe can conclude that
\nThe equation of the trend line is:
\ny = 0.20x + 15
\nIn 2025, the average fuel consumption is predicted to be about 24 mpg<\/p>\n

Convince Me!<\/strong>
\nWhy can you use a linear model to predict the y-value for a given x-value?
\nAnswer:
\nWe can use the regression line to predict values of Y was given values of X. For any given value of X, we go straight up to the line and then move horizontally to the left to find the\u00a0value of Y. The\u00a0predicted value of Y\u00a0is called the\u00a0predicted value of Y, and is denoted\u00a0Y’.<\/p>\n

Try It!<\/strong><\/p>\n

A smoothie caf\u00e9 has the ingredients needed to make 50,000 smoothies on a day when the high temperature is expected to reach 90\u00b0F. Should the caf\u00e9 employees expect to have enough ingredients for the day’s smoothie sales? Explain.
\nAnswer:<\/p>\n

KEY CONCEPT<\/strong>
\nScatter plots can be used to make predictions about current or future trends.
\nLook for the corresponding y-value for a given x-value.
\n\"Envision<\/p>\n

Find the equation of the trend line and find the y-value of a given x-value.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> How do linear models help you to make a prediction?
\nAnswer:
\nWhile\u00a0linear models\u00a0do not always accurately represent\u00a0data, and this occurs when actual\u00a0data\u00a0does not clearly show a relationship between its two variables,\u00a0linear models\u00a0are\u00a0helpful\u00a0in determining the future points of\u00a0data, the expected points of\u00a0data, and the highest possible accuracy of\u00a0data.<\/p>\n

Question 2.
\nModel with Math<\/strong>
\nHow do you find the equation of a linear model when you are given the graph but not given the equation?
\nAnswer:
\nTo simplify what has already been said, the easiest way to find the equation of a line is to look for the x and y-intercepts.
\nOne point will be (x, 0) and the other will be (0, y), where x and y are numerical values.
\nThe slope is simply
\nm = \\(\\frac{y}{x}\\).
\nWhen you have the y-intercept, (0, y),
\nyou can use the form
\ny = mx + b to find the equation for the line.
\nConsequently, with the notation used, you can represent this as
\ny=\\(\\frac{y}{x}\\)x + b
\nwhere b is the value from (0, y)
\nx is the value from (x, 0)<\/p>\n

Question 3.
\nReasoning<\/strong> Can the linear model for a set of data that is presented in a scatter plot always be used to make a prediction about any x-value? Explain.
\nAnswer:
\nYes, we can use the linear model to predict values of Y was given values of X. For any given value of X, we go straight up to the line and then move horizontally to the left to find the value of Y. The predicted value of Y is called the predicted value of Y, and is denoted Y’.<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nThe graph shows a family’s grocery expenses based on the number of children in the family,
\n\"Envision
\na. Using the slope, predict the difference in the amount spent on groceries between a family with five children and a family with two children.
\nAnswer:
\nIt is given that
\nThe graph shows a family’s grocery expenses based on the number of children in the family,
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nNow,
\nFrom the given scatter plot,
\nThe pair that represents the amount spent on groceries in a family with five children is: (5, 175)
\nThe pair that represents the amount spent on groceries in a family with two children is: (2, 140)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{140 – 175}{2 – 5}\\)
\nSlope = \\(\\frac{-35}{-3}\\)
\nSlope = \\(\\frac{35}{3}\\)
\nHence, from the above,
\nWe can conclude that using the slope, the difference in the amount spent on groceries between a family with five children and a family with two children is: \\(\\frac{35}{3}\\)<\/p>\n

b. How many children can you predict a family has if the amount spent on groceries per week is $169.47?
\nAnswer:
\nFrom the given scatter plot,
\nThe trend line equation is:
\ny = 21.08x + 85.15
\nNow,
\nIt is given that the amount spent on groceries per week is $169.47
\nSo,
\n169.47 = 21.08x + 85.15
\n21.08x = 169.47 – 85.15
\n21.08x = 84.32
\nx = \\(\\frac{84.32}{21.08}\\)
\nx = 4
\nHence, from the above,
\nWe can predict 4 children in a family if he has the amount spent on groceries per week is $169.47<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 5.
\nLeveled Practice<\/strong> The scatter plot shows the number of people at a fair based on the outside temperature. How many fewer people would be predicted to be at the fair on a 100\u00b0F day than on a 75\u00b0F day?
\nThe slope is \"Envision
\nFor each degree that the outside temperature increases, the fair attendance decreases by \"Envision thousand people.
\n\"Envision
\nThe difference between 75\u00b0F and 100\u00b0F is \"Envision \u00b0F.
\n-0.16 . \"Envision = \"Envision
\nAbout \"Envision thousand fewer people are predicted to be at the fair on a 100\u00b0F day than on a 75\u00b0F day.
\nAnswer:
\nIt is given that
\nThe scatter plot shows the number of people at a fair based on the outside temperature
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nNow,
\nWe know that,
\nThe equation of the trend line that is passing through two points is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the initial value (or) y-intercept
\nNow,
\nTo find the slope,
\nThe given points are: (75, 10K), (100, 6K)
\nWhere,
\nK is 1000
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{6K – 10K}{100 – 75}\\)
\n= –\\(\\frac{4,000}{25}\\)
\n= -160
\nSo,
\nThe equation of the trend line is:
\ny = -160x + b
\nSubstitute (100, 6K) in the above equation
\nSo,
\n6K = -160 (100) + b
\n6K + 16K = b
\nb = 22K
\nSo,
\nThe equation of the trend line is:
\ny = -160x + 22,000
\nNow,
\nAt 75\u00b0 F,
\ny = -160 (75) + 22,000
\ny = 10,000
\nAt 100\u00b0 F,
\ny = -160 (100) + 22,000
\ny = 6,000
\nHence,
\nThe difference of the people between 75\u00b0 F and 100\u00b0 F = 10,000 – 6,000
\n= 4,000
\nThe difference between 100\u00b0 F and 75\u00b0 F = 25\u00b0 F<\/p>\n

Question 6.
\nMake Sense and Persevere<\/strong> If x represents the number of years since 2000 and y represents the gas price, predict what the difference between the gas prices in 2013 and 2001 is? Round to the nearest hundredth.
\n\"Envision
\nAnswer:
\nIt is given that
\nx represents the number of years since 2000 and y represents the gas price,
\nNow,
\nWe know that,
\nThe equation of the trend line in the slope-intercept form is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the y-intercept
\nNow,
\nTo find the slope of the trend line,
\nThe given points are: (7, 3), (12, 4)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{4 – 3}{12 – 7}\\)
\nSlope = \\(\\frac{1}{5}\\)
\nSo,
\ny = \\(\\frac{1}{5}\\)x + b
\nNow,
\nSubstitute (7, 3) or (12, 4) in the above equation
\nSo,
\n3 = \\(\\frac{1}{5}\\) (7) + b
\n\\(\\frac{8}{5}\\) = b
\nSo,
\nThe equation of the trend line is:
\n5y = x + 8
\nNow,
\nThe gas prices in 2001 is:
\n5y = 1 + 8
\ny = \\(\\frac{9}{5}\\)
\ny = $1.40
\nThe gas prices in 2013 is:
\n5y = 13 + 8
\ny = \\(\\frac{21}{5}\\)
\ny = $4.20
\nSo,
\nThe difference between the gas prices in 2013 and 2001 = $4.20 – $1.40
\n= $2.80
\nHence, from the above,
\nWe can conclude that the difference between the gas prices in 2013 and 2001 is: $2.80<\/p>\n

Question 7.
\nMake Sense and Persevere<\/strong> If x represents the number of months since the beginning of 2016, and y represents the total precipitation to date, predict the amount of precipitation received between the end of March and the end of June.
\n\"Envision
\nAnswer:
\nIt is given that
\nx represents the number of months since the beginning of 2016, and y represents the total precipitation to date
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nNow,
\nFrom the given scatter plot,
\nWe can observe that the trend line starts from the origin
\nSo,
\nThe equation of the trend line that is passing through the origin is:
\ny = mx
\nwhere,
\nm is the slope
\nNow,
\nTo find the slope.
\nThe given points are: (2, 10), (10, 40)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{40 – 10}{10 – 2}\\)
\n= \\(\\frac{30}{8}\\)
\n= \\(\\frac{15}{4}\\)
\nSo,
\nThe equation of the trend line is:
\ny = \\(\\frac{15}{4}\\)x
\nNow,
\nAt the end of the march,
\nThe amount of precipitation is:
\ny = \\(\\frac{15}{4}\\) (4)
\ny = 15 in
\nAt the end of June,
\nThe amount of precipitation is:
\ny = \\(\\frac{15}{4}\\) (6)
\ny = \\(\\frac{45}{2}\\)
\ny = 22.5 inches
\nSo,
\nThe amount of precipitation between the end of March and the end of June = 22.5 – 15
\n= 7.5 in
\nHence, from the above,
\nWe can conclude that the amount of precipitation between the end of March and the end of June is: 7.5 in<\/p>\n

Question 8.
\nThe scatter plot shows a hiker’s elevation above sea level over time. The equation of the trend line shown is y = 8.77x + 686. To the nearest whole number, predict what the hiker’s elevation will be after 145 minutes.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe scatter plot shows a hiker’s elevation above sea level over time.
\nThe equation of the trend line shown is
\ny = 8.77x + 686.
\nWhere,
\n8.77 is the slope
\n686 is the initial value (or) y-intercept
\nNow,
\nFrom the graph,
\nWe can observe that
\nThe x-axis variable – Time
\nThe y-axis variable – Elevation
\nSo,
\nThe hiker’s elevation after 145 minutes is:
\ny = 8.77 (145) + 686
\ny = 1,957.65 ft
\nHence, from the above,
\nWe can conclude that the hiker’s elevation after 145 minutes will be: 1,957.65 ft<\/p>\n

Question 9.
\nMake Sense and Persevere<\/strong> The graph shows the number of gallons of water in a large tank as it is being filled. Based on the trend line, predict how long it will take to fill the tank with 375 gallons of water.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the number of gallons of water in a large tank as it is being filled
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nNow,
\nFrom the given scatter plot,
\nWe can observe that
\nThe initial value (or) y-intercept is: 15
\nNow,
\nWe know that,
\nThe equation of the trend line that has the initial value is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the y-intercept (or) initial value
\nNow,
\nTo find the slope,
\nThe required points are: (1, 30), (0, 15)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{15 – 30}{0 – 1}\\)
\n= \\(\\frac{-15}{-1}\\)
\n= 15
\nSo,
\nThe equation of the trend line is:
\ny = 15x + 15
\nNow,
\nThe time taken to fill 375 gallons of water is:
\n375 = 15x + 15
\n15x = 375 – 15
\n15x = 360
\nx = \\(\\frac{360}{15}\\)
\nx = 24 minutes
\nHence, from the above,
\nWe can conclude that the time taken to fill 375 gallons of water is: 24 minutes<\/p>\n

Question 10.
\nHigher-Order Thinking<\/strong> The graph shows the temperature, y, in a freezer x minutes after it was turned on. Five minutes after being turned on, the temperature was actually three degrees from what the trend line shows. What values could the actual temperature be after the freezer was on for five minutes?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the temperature, y, in a freezer x minutes after it was turned on. Five minutes after being turned on, the temperature was actually three degrees from what the trend line shows.
\nNow,
\nFrom the given scatter plot,
\nWe can observe that,
\nAt 5 minutes of time, the freezer temperature is 15\u00b0F
\nSo,
\nAt x = 5, y = 15
\nBut,
\nAccording to the given information
\nAt x = 5, y = 15 + 3
\nSo,
\ny = 18\u00b0F
\nHence, from the above,
\nWe can conclude that the actual temperature after the freezer was on for five minutes is: 18\u00b0F<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 11.
\nThe graph shows the altitude above sea level of a weather balloon over time. The trend line passes through the points (0, 453) and (10, 359). Which statements about the graph are true?
\n\"Envision The data show a positive correlation.
\n\"Envision The trend line is -9.4x – 453.
\n\"Envision In general, the balloon is losing altitude.
\n\"Envision The weather balloon started its flight at about 455 feet above sea level.
\n\"Envision
\n\"Envision After 4 minutes, the weather balloon had an altitude of about 415 feet above sea level.
\n\"Envision After 395 minutes, the weather balloon had an altitude of about 8 feet above sea level.
\nAnswer:
\nLet the given options be named as A, B, C, D, E and F respectively
\nIt is given that
\nThe graph shows the altitude above sea level of a weather balloon over time.
\nThe trend line passes through the points (0, 453) and (10, 359)
\nWe know that,
\nThe equation of the trend line that is passing through two points is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the initial value (or) y-intercept
\nWe know that
\nThe “y-intercept” is the value of y when x= 0
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{359 – 453}{10 – 0}\\)
\n= \\(\\frac{-94}{10}\\)
\n= -9.4
\nSo,
\nThe equation of the trend line is:
\ny = -9.4x + 453
\nHence, from the above,
\nWe can conclude that options C, D, and E matches the given situation<\/p>\n

Topic 4 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary<\/strong> How can you use a trend line to determine the type of linear association for a scatter plot? Lesson 4.2
\nAnswer:
\nThe straight\u00a0line\u00a0is a\u00a0trend line, designed to come as close as possible to all the data points. The\u00a0trend line\u00a0has a positive slope, which shows a positive\u00a0relationship\u00a0between X and Y. The points in the\u00a0graph\u00a0are tightly clustered about the\u00a0trend line\u00a0due to the strength of the\u00a0relationship\u00a0between X and Y.<\/p>\n

The scatter plot shows the amount of time Adam spent studying and his test scores. Use the scatter plot for Items 2-4.<\/p>\n

Question 2.
\nWhat relationship do you see between the amount of time spent studying and the test scores? Is the relationship linear? Lesson 4.1
\nA. In general, Adam scores higher on a test when he spends more time studying. There is not a linear relationship.
\nB. In general, Adam scores higher on a test when he spends more time studying. There is a positive linear relationship.
\n\"Envision
\nC. In general, Adam scores lower on a test when he spends more time studying. There is a negative linear relationship.
\nD. In general, Adam scores lower on a test when he spends more time studying. There is no relationship.
\nAnswer:
\nIt is given that
\nThe scatter plot shows the amount of time Adam spent studying and his test scores
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that
\nThe association or correlation is positive and there is a linear relationship
\nAdam is scoring higher on a test when he is studying for more hours
\nHence, from the above,
\nWe can conclude that option B matches the given situation<\/p>\n

Question 3.
\nUse the y-intercept and the point (4,90) from the line on the scatter plot. What is the equation of the linear model? Lesson 4.3
\nAnswer:
\nThe given scatter plot is (From Question 2):
\n\"Envision
\nNow,
\nFrom the given scatter plot,
\nWe can observe that
\nThe initial value (or) y-intercept is: 60
\nWe know that,
\nThe “y-intercept” is the value of y when x = 0
\nSo,
\nThe points required to find the equation of the scatter plot is: (0, 60), (4, 90)
\nWe know that,
\nThe equation of the trend line that has y-intercept is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the initial value (or) y-intercept
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{90 – 60}{4 – 0}\\)
\n= \\(\\frac{30}{4}\\)
\n= \\(\\frac{15}{2}\\)
\nSo,
\nThe equation of the trend line is:
\ny = \\(\\frac{15}{2}\\)x + 60
\nHence, from the above,
\nWe can conclude that the equation of the linear line is:
\ny = \\(\\frac{15}{2}\\)x + 60<\/p>\n

Question 4.
\nPredict Adam’s test score when he studies for 6 hours. Lesson 4.3
\nAnswer:
\nWe know that,
\nFrom the given scatter plot,
\nThe variable on the x-axis is: Time
\nThe variable on the y-axis is: Test scores
\nNow,
\nFrom Problem 3,
\nThe equation of the trend line is:
\ny = \\(\\frac{15}{2}\\)x + 60
\nAt 6 hours,
\ny = \\(\\frac{15}{2}\\) (6) + 60
\ny = 105
\nHence, from the above,
\nWe can conclude that Adam’s test score is 105 when he studies for 6 hours<\/p>\n

Question 5.
\nDescribe the relationship between the data in the scatter plot. Lesson 4.2
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the above scatter plot,
\nWe can observe that as the value of x increases, the value of y decreases
\nHence, from the above,
\nWe can conclude that the given scatter plot has the negative association<\/p>\n

Question 6.
\nThe scatter plot shows the mean annual temperature at different elevations. Select all the observations that are true about the scatter plot. Lesson 4.1
\n\"Envision The majority of the elevations are in a cluster between 1,250 meters and 2,250 meters.
\n\"Envision There is a gap in the data between 500 meters and 1,250 meters.
\n\"Envision
\n\"Envision There is an outlier at about (50, 21).
\n\"Envision In general, the mean annual temperature decreases as the elevation increases.
\n\"Envision Because there is a gap in the values, there is no association between the temperature and elevation.
\nAnswer:
\nLet the given options be named as A, B, C, D, and E
\nIt is given that
\nThe scatter plot shows the mean annual temperature at different elevations
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that
\nThere is a cluster between 1,250 m and 2,250m
\nThere is a gap between 500m and 1,250m
\nIn general, the mean annual temperature decreases as the elevation increases.
\nBecause there is a gap in the values, there is no association between the temperature and elevation.
\nHence, from the above,
\nWe can conclude that A, B, D, and E matches the given situation
\nHow well did you do on the mid-topic checkpoint? Fill in the stars. \"Envision<\/p>\n

Topic 4 MID-TOPIC PERFORMANCE TASK<\/h3>\n

A pitcher’s ERA (earned run average) is the average number of earned runs the pitcher allows every 9 innings pitched. The table shows the ERA and the number of wins for starting pitchers in a baseball league.
\n\"Envision<\/p>\n

PART A<\/strong>
\nConstruct a scatter plot of the data in the table.
\nAnswer:
\nIt is given that
\nA pitcher’s ERA (earned run average) is the average number of earned runs the pitcher allows every 9 innings pitched. The table shows the ERA and the number of wins for starting pitchers in a baseball league.
\nNow,
\nThe given table is:
\n\"Envision
\nHence,
\nThe representation of the scatter plot for the given data is:
\n\"\"<\/p>\n

PART B<\/strong>
\nIdentify the association between the data. Explain the relationship between ERA and the number of wins shown in the scatter plot.
\nAnswer:
\nFrom the above scatter plot,
\nWe can observe that
\nAs the value of x increases, the value of y decreases
\nHence, from the above,
\nWe can conclude that the relationship between ERA and the number of wins as shown in the above scatter plot is a “Negative Correlation”<\/p>\n

PART C<\/strong>
\nDraw a trend line. Write an equation of the linear model. Predict the number of wins of a pitcher with an ERA of 6.
\nAnswer:
\nWe know that,
\nThe equation of the trend line between two points is:
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the initial value (or) y-intercept
\nNow,
\nTo find the slope,
\nThe points are: (5, 4), (2, 10)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nNow,
\nWe know that,
\nSlope = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope = \\(\\frac{10 – 4}{2 – 5}\\)
\n= –\\(\\frac{6}{3}\\)
\n= -2
\nSo,
\nThe equation of the trend line is:
\ny = -2x + b
\nNow,
\nSubstitute (5, 4) in the above equation
\nSo,
\n4 = -2 (5) + b
\n4 + 10 = b
\nb = 14
\nSo,
\nThe equation of the trend line is:
\ny = -2x + 14
\nNow,
\nFrom the given table,
\nWe can observe that
\nThe x-axis variable is: ERA
\nThe y-axis variable is: The number of wins
\nSo,
\nFor x = 6,
\ny = -2 (6) + 14
\ny = 14 – 12
\ny = 2
\nHence, from the above,
\nWe can conclude that the number of wins of a pitcher with an ERA of 6 is: 2<\/p>\n

Lesson 4.4 Interpret Two-Way Frequency Tables<\/h3>\n

Explore It!<\/strong>
\nThe owners of a ski resort want to know which is more popular, skiing or snowboarding. The resort conducts a poll, asking visitors their age and which activity they prefer. The results are shown in the table.
\n\"Envision<\/p>\n

A. Use the table to describe the visitors polled.
\nAnswer:
\nIt is given that
\nThe owners of a ski resort want to know which is more popular, skiing or snowboarding. The resort conducts a poll, asking visitors their age and which activity they prefer. The results are shown in the table.
\nNow,
\nFrom the given table,
\nWe can observe that the visiters polled are of the age below 35 and above 35
\nThe activities for which the visitors polled are: Skiing, Snowboarding<\/p>\n

B. What information can the owners of the resort determine from the data in the table?
\nAnswer:
\nFrom the given table,
\nWe can observe that
\nThe number of visitors that had polled for Skiing and snowboarding
\nHence, from the above,
\nWe can conclude that the information the owners of the resort determine from the data in the table is the number of visitors that had polled for Skiing and Snowboarding<\/p>\n

C. Make a statement that is supported by the data.
\nAnswer:
\nThe statement that is supported by the given data is:
\nThe number of visitors that are over 35 years is the majority who polled for Skiing<\/p>\n

Focus on math practices<\/strong>
\nModel with Math
\n<\/strong> How else might you display the data to show the relationship between people’s ages and which activity they prefer?
\nAnswer:
\nThe other way to display the data to show the relationship between people’s ages and the activity they prefer is:
\n\"\"<\/p>\n

Essential Question<\/strong>
\nHow does a two-way frequency table show the relationships between sets of paired data?
\nAnswer:
\nTwo-way frequency tables are\u00a0a visual representation of the possible\u00a0relationships between two sets\u00a0of\u00a0categorical data. The categories\u00a0are\u00a0labeled at the top and the left side of the\u00a0table,\u00a0with the frequency\u00a0(count) information appearing in the four (or more) interior cells of the\u00a0table.<\/p>\n

Try It!<\/strong>
\nA weatherman asks 75 people from two different cities if they own rain boots. Complete the two-way frequency table to show the results of the survey.
\n\"Envision
\nAnswer:
\nIt is given that
\nA weatherman asks 75 people from two different cities if they own rain boots.
\nSo,
\nThe total number of people who say whether they own rain boots or not are: 75
\nSo,
\n(The people who say yes + The people who say no)City A<\/sub> + (The people who say yes + the people who say no)City B<\/sub> = 75
\nHence,
\nThe complete two-way frequency table that shows the results of the survey is:
\n\"\"<\/p>\n

Convince Me!<\/strong>
\nWhat pattern do you see in the two-way frequency table?
\nAnswer:
\nTwo-way frequency tables are\u00a0a visual representation of the possible relationships between\u00a0two\u00a0sets of categorical data. The categories\u00a0are\u00a0labeled at the top and the left side of the\u00a0table, with the\u00a0frequency\u00a0(count) information appearing in the four (or more) interior cells of the\u00a0table.<\/p>\n

Try It!<\/strong>
\nOne hundred students were asked how they traveled to school. Of the girls, 19 rode in a car, 7 rode the bus, and 27 took the train. Of the boys, 12 took the train, 25 rode in a car, and 10 rode the bus. Construct a two-way frequency table. Then tell which mode of transportation is the most popular. Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nOne hundred students were asked how they traveled to school. Of the girls, 19 rode in a car, 7 rode the bus, and 27 took the train. Of the boys, 12 took the train, 25 rode in a car, and 10 rode the bus
\nSo,
\nThe complete two-way frequency table for the given situation is:
\n\"\"
\nNow,
\nFrom the above, two-way frequency table,
\nWe can observe that more number of people preferred car mode of transportation
\nHence, from the above,
\nWe can conclude that the car mode of transportation is the most popular<\/p>\n

KEY CONCEPT<\/strong>
\nA two-way frequency table displays the relationship between paired categorical data. You can interpret the data in the table to draw conclusions.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> How does a two-way frequency table show the relationship between sets of paired categorical data?
\nAnswer:
\nTwo-way frequency tables are\u00a0a visual representation of the possible\u00a0relationships between two sets\u00a0of\u00a0categorical data. The categories\u00a0are\u00a0labeled at the top and the left side of the\u00a0table,\u00a0with the frequency\u00a0(count) information appearing in the four (or more) interior cells of the\u00a0table.<\/p>\n

Question 2.
\nModel with Math<\/strong>
\nHow do you decide where to start filling in a two-way frequency table when some of the data are already there?
\nAnswer:
\nIn a two-way frequency table, when there is already data present,
\nYou have to start filling in where there is more data in the table so that all the frequencies can be counted easily and filling of the frequency table will also be fast<\/p>\n

Question 3.
\nUse Structure<\/strong> How can you use the structure of a two-way frequency table to complete it?
\nAnswer:
\nThe steps that are used to complete the two-way frequency table is:
\nStep 1:
\nIdentify the variables. There are two variables of interest here: the commercial viewed and opinion
\nStep 2:
\nDetermine the possible values of each variable. For the two variables, we can identify the following possible values
\nStep 3:
\nSet up the table
\nStep 4:
\nFill in the frequencies<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nA basketball coach closely watches the shots of 60 players during basketball tryouts. Complete the two-way frequency table to show her observations.
\n\"Envision
\nAnswer:
\nIt is given that
\nA basketball coach closely watches the shots of 60 players during basketball tryouts.
\nSo,
\nUnderclassmen + Upperclassmen = 60
\nHence,
\nThe complete two-way frequency table that shows the basketball coach’s observations is:
\n\"\"<\/p>\n

Question 5.
\nDo the data in the two-way frequency table support the following statement? Explain.
\nThere are more middle school students who wear glasses than high school students who wear contacts.
\n\"Envision
\nAnswer:
\nThe given statement is:
\nThere are more middle school students who wear glasses than high school students who wear contacts.
\nNow,
\nThe given two-way frequency table is:
\n\"Envision
\nFrom the given two-way frequency table,
\nThe number of middle school students who wear glasses is: 13
\nThe number of high school students who wear contacts is: 20
\nSo,
\nThe number of middle school students who wear glasses < The number of high school students who wear contacts
\nHence, from the above,
\nWe can conclude that the given statement can’t be supported<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice in 6-8, complete the two-way frequency tables.<\/p>\n

Question 6.
\nYou ask 70 of your classmates if they have any siblings. Complete the two-way frequency table to show the results of the survey.
\n\"Envision
\nAnswer:
\nIt is given that
\nYou ask 70 of your classmates if they have any siblings
\nSo,
\n(The number of boys and girls who have siblings) + (The number of girls who do not have siblings) = 70
\nHence,
\nThe complete two-way frequency table that shows the survey results is:
\n\"\"<\/p>\n

Question 7.
\nA company surveyed 200 people and asked which car model they preferred. Complete the two-way frequency table to show the results of the survey.
\n\"Envision
\nAnswer:
\nIt is given that
\nA company surveyed 200 people and asked which car model they preferred
\nSo,
\n(The number of males and females who preferred 2-door car model) + (The number of males and females who preferred 4-door car model) = 200
\nHence,
\nThe complete two-way frequency table that shows the results of the survey is:
\n\"\"<\/p>\n

Question 8.
\nMake Sense and Persevere<\/strong>
\nYou ask 203 of your classmates how they feel about the school year being made longer. Complete the two-way frequency table to show the results of the survey.
\n\"Envision
\nAnswer:
\nIt is given that
\nYou ask 203 of your classmates how they feel about the school year being made longer
\nSo,
\n6th Grade students + 7th Grade students + 8th Grade students = 203
\nHence,
\nThe complete two-way frequency table that shows the complete survey results is:
\n\"\"<\/p>\n

Question 9.
\nStudents at a local school were asked, “About how many hours do you spend on homework each week?” The two-way frequency table shows the results of the survey. Classify the statement below as true or false. Explain.
\nMore students study for 5 to 6 hours than for 1 to 2 hours.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe two-way frequency table shows the results of the survey.
\nNow,
\nThe given statement is:
\nMore students study for 5 to 6 hours than for 1 to 2 hours.
\nNow,
\nThe given two-way frequency table is:
\n\"Envision
\nFrom the given two-way frequency table,
\nWe can observe that
\nThe number of students who study for 5 – 6 hours is: 104
\nThe number of students who study for 1 – 2 hours is: 147
\nSo,
\nThe number of students who study for 1 – 2 hours > The number of students who study for 5 – 6 hours
\nHence, from the above,
\nWe can conclude that the given statement is false<\/p>\n

Question 10.
\nHigher-Order Thinking<\/strong> Demi and Margaret record the weather in their respective cities on weekend days over the summer. a. Construct a single, two-way frequency table to show the results.
\n\"Envision
\n\"Envision
\nAnswer:
\nIt is given that
\nDemi and Margaret record the weather in their respective cities on weekend days over the summer
\nNow,
\nThe given information regarding the given situation is:
\n\"Envision
\nLet the struck lines be the number of times that have no rain
\nLet the non-struck lines be the number of lines that have rain
\nHence,
\nThe complete two-way frequency table that shows the results is:
\n\"\"<\/p>\n

b. Which day saw the least rain? Explain.
\nAnswer:
\nThe condition for the least rain is: The number of times rain occurs + The number of times that no rain occurs
\nHence, from the above,
\nWe can conclude that Saturday saw the least rain<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 11.
\nAt one point last year, the local animal shelter had only cats and dogs. There were 74 animals in all. Of the cats, 25 were male and 14 were female. Of the dogs, 23 were male and 12 were female.
\nPART A<\/strong>
\nConstruct a two-way frequency table of
\n\"Envision
\nAnswer:
\nIt is given that
\nAt one point last year, the local animal shelter had only cats and dogs. There were 74 animals in all. Of the cats, 25 were male and 14 were female. Of the dogs, 23 were male and 12 were female.
\nHence,
\nThe complete two-way table that shows the survey results is:
\n\"\"
\nPART B<\/strong>
\nFor which gender, male or female, is there the data.
\na greater need for pet adoption? Explain.
\nA. There are almost twice as many female pets, so there is a greater need for people to adopt female dogs and cats.
\nB. There are almost twice as many male pets, so there is a greater need for people to adopt male dogs and cats.
\nC. There are almost twice as many female pets, so there is a greater need for people to adopt male dogs and cats.
\nD. There are almost twice as many male pets, so there is a greater need for people to adopt female dogs and cats.
\nAnswer:
\nFrom part (A),
\nthe two-way frequency table that matches the given situation is:
\n\"\"
\nSo,
\nFrom the above two-way frequency table,
\nWe can observe that
\nThere are almost twice as males as females
\nHence, from the above,
\nWe can conclude that option D matches the given situation perfectly<\/p>\n

Lesson 4.5 Interpret TwoWay Relative Frequency Tables<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Mr. Day’s math class asked 200 cell phone owners which size phone they prefer. They presented the results in a two-way frequency table. How can you use the data to compare the percent of students who chose the small screen to the percent of adults who chose the small screen?
\n\"Envision
\nAnswer:
\nIt is given that
\nMr. Day’s math class asked 200 cell phone owners which size phone they prefer. They presented the results in a two-way frequency table
\nNow,
\nFrom the given two-way frequency table,
\nWe can observe that
\nThe number of students who chose the small screen is: 48
\nThe number of adults who chose the small screen is: 18
\nNow,
\nThe percent of students who chose the small screen = \\(\\frac{The number of small screens chosen by the students}{The total number of screens}\\) \u00d7 100
\n= \\(\\frac{48}{200}\\) \u00d7 100
\n= 24%
\nThe percent of adults who chose the small screen = \\(\\frac{The number of small screens chosen by the adults}{The total number of screens}\\) \u00d7 100
\n= \\(\\frac{18}{200}\\) \u00d7 100
\n= 9%
\nSo,
\nThe percent of students who chose the small screen to the percent of adults who chose the small screen
\n= \\(\\frac{9}{24}\\) \u00d7 100
\n= \\(\\frac{9 \u00d7 100}{24}\\)
\n= 37.5%
\nHence, from the above,
\nWe can conclude that the percent of students who chose the small screen to the percent of adults who chose the small screen is: 37.5%<\/p>\n

Make Sense and Persevere<\/strong>
\nHow do two-way frequency tables allow you to interpret relationships between categorical data using rows and columns?
\nAnswer:
\nTwo-way frequency tables are\u00a0a visual representation of the possible\u00a0relationships between two sets\u00a0of\u00a0categorical data. The categories\u00a0are\u00a0labeled at the top and the left side of the\u00a0table,\u00a0with the frequency\u00a0(count) information appearing in the four (or more) interior cells of the\u00a0table.<\/p>\n

Focus on math practices<\/strong>
\nMake Sense and Persevere How does know a percentage change the way you interpret the results?
\nAnswer:
\nFirst: work out the\u00a0difference\u00a0(increase) between the two numbers you are comparing. Then: divide the\u00a0increase\u00a0by the original number and multiply the answer by 100.
\nSo,
\n%\u00a0increase\u00a0=\u00a0Increase\u00a0\u00f7 Original Number \u00d7 100.
\nIf your answer is a negative number, then this is a\u00a0percentage\u00a0decrease.<\/p>\n

? Essential Question<\/strong>
\nWhat is the advantage of a two-way relative frequency table for showing relationships between sets of paired data?
\nAnswer:
\nTwo-way relative frequency tables show\u00a0us percentages rather than counts. They are good for seeing if there is an association\u00a0between two\u00a0variables<\/p>\n

Try It!<\/strong>
\nAsha asked 82 classmates whether they play sports on the weekend. The results are shown in the two-way frequency table below.
\n\"Envision<\/p>\n

Convince Me!<\/strong>
\nHow is a two-way relative frequency table different from a two-way frequency table?
\nAnswer:
\nWhen a\u00a0two-way table\u00a0displays percentages or ratios (called\u00a0relative frequencies), instead of just\u00a0frequency\u00a0counts, the\u00a0table\u00a0is referred to as a\u00a0two-way relative frequency table. These\u00a0two-way tables\u00a0can show\u00a0relative frequencies\u00a0for the whole\u00a0table, for rows, or for columns.<\/p>\n

Use Asha’s two-way frequency table to complete the two-way relative frequency table.
\n\"Envision
\nAnswer:
\nThe given two-way table is:
\n\"Envision
\nNow,
\nWe know that,
\nThe % of boys or girls who say yes = \\(\\frac{The number of boys or girls who say yes}{The total number of people}\\) \u00d7 100
\nThe % of boys or girls who say no = \\(\\frac{The number of boys or girls who say no}{The total number of people}\\) \u00d7 100
\nFrom the given two-way frequency table,
\nThe total number of people is: 82
\nHence,
\nThe complete two-way relative frequency table for the given situation is:
\n\"\"<\/p>\n

Try It!<\/strong>
\nUse the data in the table below.
\n\"\"
\na. How does the percent of students who choose e-books compare to the percent of students who choose audiobooks?
\nAnswer:
\nFrom the given two-way relative frequency table,
\nWe can observe that
\nThe % of students who choose e-books is: 52%
\nThe % of students who choose Audiobooks is: 48%
\nSo,
\nThe % of students who choose e-books to the % of students who choose audiobooks
\n= \\(\\frac{48}{52}\\) \u00d7 100
\n= 92.3%
\nHence, from the above,
\nWe can conclude that the % of students who choose e-books to the % of students who choose audiobooks is: 92.3%<\/p>\n

b. Is there evidence that 7th graders have a greater tendency to choose audiobooks? Explain.
\nAnswer:
\nFrom the given two-way relative frequency table,
\nThe % of 7th-grade students who choose audiobooks is: 58.9%
\nThe % of the 6th-grade students who choose audiobooks is: 36.5%
\nSo,
\nThe % of 7th-grade students who choose audiobooks > The % of 6th-grade students who choose audiobooks
\nHence, from the above,
\nWe can conclude that there is a piece of evidence that 7th-graders have a greater tendency to choose audiobooks<\/p>\n

KEY CONCEPT<\/strong>
\nRelative frequency is the ratio of a data value to the total of a row, a column, or the entire data set. It is expressed as a percent. A total two-way relative frequency table gives the percent of the population that is in each group.
\nIn a row two-way relative frequency table, the percents in each row add up to 100%.
\nIn a column two-way relative frequency table, the percents in each column add up to 100%.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong>
\nWhat is the advantage of a two-way relative frequency table for showing relationships between sets of paired data?
\nAnswer:
\nTwo-way relative frequency tables show\u00a0us percentages rather than counts. They are good for seeing if there is an association\u00a0between two\u00a0variables<\/p>\n

Question 2.
\nReasoning<\/strong> when comparing relative frequency by rows or columns only, why do the percentages not total 100%? Explain.
\nAnswer:
\nWhen comparing relative frequency by rows or columns only, the individual percentages will not be 100%
\nSo, their total will also not be equal to 100%<\/p>\n

Question 3.
\nCritique Reasoning<\/strong>
\nMaryann says that if 100 people are surveyed, the frequency table will provide the same information as a total relative frequency table. Do you agree? Explain why or why not.
\nAnswer:
\nIt is given that
\nMaryann says that if 100 people are surveyed, the frequency table will provide the same information as a total relative frequency table
\nWe know that,
\nThe “Two-way frequency table” gives us information about the categories in the form of counts and frequencies
\nThe “Two-way relative frequency table” gives us information about the categories in terms of percentages of frequencies
\nHence, from the above,
\nWe can agree with Maryann<\/p>\n

Do You Know How?<\/p>\n

In 4-6, use the table. Round to the nearest percent.
\n\"Envision<\/p>\n

Question 4.
\nWhat percent of the people surveyed have the artistic ability?
\nAnswer:
\nFrom the table,
\nThe number of people who have the artistic ability is: 101
\nNow,
\nWe know that,
\nThe % of the people surveyed that have the artistic ability = \\(\\frac{The total number of people who have the artistic ability}{The total number of people}\\) \u00d7 100
\n= \\(\\frac{101}{223}\\) \u00d7 100
\n= 45%
\nHence, from the above,
\nWe can conclude that 45% of the people surveyed have the artistic ability<\/p>\n

Question 5.
\nWhat percent of left-handed people surveyed have the artistic ability?
\nAnswer:
\nFrom the table,
\nThe number of left-handed people who have artistic ability is: 86
\nNow,
\nWe know that,
\nThe % of the left-handed people surveyed that have the artistic ability = \\(\\frac{The total number of left-handed people who have the artistic ability}{The total number of people}\\) \u00d7 100
\n= \\(\\frac{86}{223}\\) \u00d7 100
\n= 39%
\nHence, from the above,
\nWe can conclude that 39% of the left-handed people surveyed have the artistic ability<\/p>\n

Question 6.
\nWhat percent of the people who have the artistic ability are left-handed?
\nAnswer:
\nFrom the table,
\nThe number of left-handed people who have artistic ability is: 86
\nNow,
\nWe know that,
\nThe % of the left-handed people surveyed that have the artistic ability = \\(\\frac{The total number of left-handed people who have the artistic ability}{The total number of people}\\) \u00d7 100
\n= \\(\\frac{86}{223}\\) \u00d7 100
\n= 39%
\nHence, from the above,
\nWe can conclude that 39% of the left-handed people surveyed have the artistic ability<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice in 7-8, complete the two-way relative frequency tables.<\/p>\n

Question 7.
\nIn a group of 120 people, each person has a dog, a cat, or a bird. The two-way frequency table shows how many people have each kind of pet. Complete the two-way relative frequency table to show the distribution of the data with respect to all 120 people. Round to the nearest tenth of a percent.
\n\"Envision
\nAnswer:
\nIt is given that
\nIn a group of 120 people, each person has a dog, a cat, or a bird. The two-way frequency table shows how many people have each kind of pet
\nHence,
\nThe complete two-way relative frequency table for the given situation is:
\n\"\"<\/p>\n

Question 8.
\nThere are 55 vehicles in a parking lot. The two-way frequency table shows data about the types and colors of the vehicles. Complete the two-way relative frequency table to show the distribution of the data with respect to color. Round to the nearest tenth of a percent.
\n\"Envision
\nAnswer:
\nIt is given that
\nThere are 55 vehicles in a parking lot. The two-way frequency table shows data about the types and colors of the vehicles.
\nHence,
\nThe completed two-way relative frequency table that shows the distribution of the data with respect to color is:
\n\"\"<\/p>\n

Question 9.
\nMen and women are asked what type of car they own. The table shows the relative frequencies with respect to the total population asked. Which type of car is more popular?
\n\"Envision
\n\"Envision
\nAnswer:
\nIt is given that
\nMen and women are asked what type of car they own. The table shows the relative frequencies with respect to the total population asked.
\nNow,
\nThe given two-way relative frequency table is:
\n\"Envision
\nNow,
\nFrom the given table,
\nWe can observe that
\nMost of the people have shown interest in the 4-door type of car
\nHence, from the above,
\nWe can conclude that the 4-door type of car is more popular<\/p>\n

Question 10.
\nMake Sense and Persevere<\/strong> Students were asked if they like raspberries. The two-way relative frequency table shows the relative frequencies with respect to the response.
\n\"Envision
\na. What percent of students who do not like raspberries are girls?
\nAnswer:
\nIt is given that
\nStudents were asked if they like raspberries. The two-way relative frequency table shows the relative frequencies with respect to the response.
\nNow,
\nFrom the given two-way frequency table,
\nWe can observe that the % of girls who do not like raspberries are: 48%
\nHence, from the above,
\nWe can conclude that the % of students who do not like raspberries are girls is: 48%<\/p>\n

b. Is there evidence of an association between the response and the gender? Explain.
\nAnswer:
\nFrom the given two-way relative frequency table,
\nWe can observe that
\nThe % of girls who like raspberries is more than the % of boys who like raspberries
\nThe % of girls who do not like raspberries is less than the % of boys who do not like raspberries<\/p>\n

Question 11.
\nHigher-Order Thinking<\/strong> All the workers in a company were asked a survey question. The two-way frequency table shows the responses from the workers in the day shift and night shift.
\n\"Envision
\na. Construct a two-way relative frequency table to show the relative frequencies with respect to the shift.
\n\"Envision
\nAnswer:
\nIt is given that
\nAll the workers in a company were asked a survey question. The two-way frequency table shows the responses from the workers in the day shift and night shift.
\nNow,
\nThe given two-way frequency table is:
\n\"Envision
\nHence,
\nThe completed two-way relative frequency table for the survey is:
\n\"\"<\/p>\n

b. Is there evidence of an association between the response and the shift? Explain.
\nAnswer:
\nFrom the two-way frequency table that is mentioned in part (a),
\nWe can observe that
\nThe % of people who opted for the day shift are more than the % of people who opted for the night shift<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 12.
\nPatients in a blind study were given either Medicine A or Medicine B. The table shows the relative frequencies
\n\"Envision
\nIs there evidence that improvement was related to the type of medicine? Explain.
\nA. The same number of people took each medicine, but the percent of people who reported improvement after taking Medicine B was significantly greater than the percent for Medicine A.
\nB. The same number of people took each medicine, but the percent of people who reported
\nimprovement after taking Medicine A was significantly greater than the percent for Medicine B.
\nC. Different numbers of people took each medicine, but the percent of people who reported improvement after taking Medicine B was significantly greater than the percent for Medicine A.
\nD. Different numbers of people took each medicine, but the percent of people who reported improvement after taking Medicine A was significantly greater than the percent for Medicine B.
\nAnswer:
\nIt is given that
\nPatients in a blind study were given either Medicine A or Medicine B. The table shows the relative frequencies
\nWe know that,
\nThe number of people will be different
\nNow,
\nWhen we observe the given two-way related frequency table,
\nThe improvement due to Medicine B > The improvement due to Medicine A
\nHence, from the above,
\nWe can conclude that option C matches the given situation<\/p>\n

3-Act Mathematical Modeling: Reach Out<\/h3>\n

3-ACT MATH<\/p>\n

\"Envision<\/p>\n

ACT 1<\/strong><\/p>\n

Question 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\nAnswer:<\/p>\n

Question 3.
\nConstruct Arguments Predict an answer to this Main Question. Explain your prediction.
\n\"Envision
\nAnswer:<\/p>\n

Question 4.
\nOn the number line below, write a number that is too small to be the answer. Write a number that is too large.
\n\"Envision
\nAnswer:<\/p>\n

Question 5.
\nPlot your prediction on the same number line.
\nAnswer:<\/p>\n

ACT 2<\/strong><\/p>\n

Question 6.
\nWhat information in this situation would be helpful to know? How would you use that information?
\n\"Envision
\nAnswer:<\/p>\n

Question 7.
\nUse Appropriate Tools<\/strong> What tools can you use to solve the problem? Explain how you would use them strategically.
\nAnswer:<\/p>\n

Question 8.
\nModel with Math<\/strong>
\nRepresent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 9.
\nWhat is your answer to the Main Question? Is it higher or lower than your initial prediction? Explain why.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong><\/p>\n

Question 10.
\nWrite the answer you saw in the video.
\nAnswer:<\/p>\n

Question 11.
\nReasoning<\/strong> Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\nAnswer:<\/p>\n

Question 12.
\nMake Sense and Persevere<\/strong> Would you change your model now that you know the answer? Explain.
\n\"Envision
\nAnswer:<\/p>\n

Reflect<\/strong><\/p>\n

Question 13.
\nModel with Math<\/strong>
\nExplain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\n\"Envision
\nAnswer:<\/p>\n

Question 14.
\nCritique Reasoning<\/strong> Choose a classmate’s model. How would you adjust that model?
\nAnswer:<\/p>\n

SEQUEL<\/strong><\/p>\n

Question 15.
\nModel with Math<\/strong> Measure a classmate’s wingspan. Use your model to predict your classmate’s height. How well did your model predicts your classmate’s actual height?
\n\"Envision
\nAnswer:<\/p>\n

Topic 4 REVIEW<\/h3>\n

? Topic Essential Question<\/strong><\/p>\n

How can you represent the relationship between paired data and use the representation to make predictions?
\nAnswer:
\nThe\u00a0scatter\u00a0diagram graphs\u00a0pairs of\u00a0numerical\u00a0data, with one variable on each axis,\u00a0to\u00a0look for a\u00a0relationship between\u00a0them. If the variables\u00a0are\u00a0correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line<\/p>\n

Vocabulary Review<\/strong><\/p>\n

Match each example on the left with the correct word and then provide another example.<\/p>\n

\"Envision<\/p>\n

\"Envision
\nAnswer:
\n\"\"<\/p>\n

Use Vocabulary in Writing<\/strong>
\nDescribe the scatter plot at the right. Use vocabulary terms in your description.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nNow,
\nFrom the given scatter plot,
\nWe can observe that
\nWith the help of measurement data, a trend line is drawn
\nThere is a trend line that is not passing through the origin
\nThere is an outlier present in the given scatter plot<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

LESSON 4.1 Construct and Interpret Scatter Plots<\/strong><\/p>\n

Quick Review<\/strong>
\nA scatter plot shows the relationship between paired measurement data. Scatter plots can be used to interpret data by looking for clusters, gaps, and outliers.<\/p>\n

Practice<\/strong>
\nThe table shows the distance in miles and the price of airfare in dollars.
\n\"Envision<\/p>\n

Question 1.
\nConstruct a scatter plot.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe table shows the distance in miles and the price of airfare in dollars.
\nHence,
\nThe representation of the scatter plot that describes the relationship between the price of airfare and distance is:
\n\"\"<\/p>\n

Question 2.
\nIs there a relationship between distance and airfare? Explain.
\nAnswer:
\nFrom the above scatter plot,
\nWe can observe that there is no association between distance and airfare
\nHence, from the above,
\nWe can conclude that there is no relationship between distance and airfare<\/p>\n

LESSON 4.2 Analyze Linear Associations<\/strong><\/p>\n

Quick Review<\/strong>
\nThe association between the data in a scatter plot can be linear or nonlinear. A trend line is a line on a scatter plot, drawn near the points, which approximates the association between paired data. If the data are linear, the association can be positive or negative, and strong or weak.<\/p>\n

Practice<\/strong>
\nIdentify the association between the data on each scatter plot.<\/p>\n

Question 1.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nNow,
\nFrom the given scatter plot,
\nWe can observe that as the value of x increases, the value of y decreases
\nHence, from the above,
\nWe can conclude that there is a negative association between the data in the scatter plot<\/p>\n

Question 2.
\n\"Envision
\nAnswer:
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that the data is in a non-linear trend
\nHence, from the above,
\nWe can conclude that there is a non-linear association between the data in the scatter plot<\/p>\n

LESSON 4.3 Use Linear Models to Make Predictions<\/strong><\/p>\n

Quick Review<\/strong>
\nTo make predictions, substitute known values into the equation of a linear model to solve for an unknown.<\/p>\n

Practice<\/strong>
\nThe scatter plot shows the wages of employees.
\n\"Envision<\/p>\n

Question 1.
\nIf an employee earns $570, what is the expected number of copies sold?
\nAnswer:
\nIt is given that
\nThe scatter plot shows the wages of employees.
\nNow,
\nThe given scatter plot is:
\n\"Envision
\nFrom the given scatter plot,
\nWe can observe that
\nThe equation of the trend line is:
\ny = 6x + 120
\nWhere,
\ny is the wages
\nx is the number of copies sold
\nNow,
\nFor y = $570,
\n570 = 6x + 120
\n6x = 570 – 120
\n6x = 450
\nx = \\(\\frac{450}{6}\\)
\nx = 75
\nHence, from the above,
\nWe can conclude that for an employee wage of $570, the number of copies sold is: 75<\/p>\n

Question 2.
\nIf an employee sells 100 copies, what is the expected wage?
\nAnswer:
\nFor x = 100,
\ny = 6x + 120
\ny = 6 (100) + 120
\ny = 600 + 120
\ny = 720
\nHence,f rom the above,
\nWe can conclude that
\nIf an employee sells 100 copies, then the expected wage is: $720<\/p>\n

LESSON 4.4 Interpret Two-Way Frequency Tables<\/strong><\/p>\n

Quick Review<\/strong>
\nA two-way frequency table displays the relationship between paired categorical data.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nThe two-way frequency table shows the results of a random survey of movies watched by 100 students. Mrs. Leary said that according to the data, girls are more likely than boys to watch movie A. Is the statement true or false? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe two-way frequency table shows the results of a random survey of movies watched by 100 students.
\nNow,
\nThe given statement is:
\nMrs. Leary said that according to the data, girls are more likely than boys to watch movie A.
\nNow,
\nThe given two-way frequency table is:
\n\"Envision
\nNow,
\nFrom the given two-way frequency table,
\nWe can observe that
\nThe number of girls who watch movie A > The number of boys who watch movie A
\nHence, from the above,
\nWe can conclude that the given statement is true<\/p>\n

LESSON 4.5 Interpret Two-Way Relative Frequency Tables<\/strong><\/p>\n

Quick Review<\/strong>
\nRelative frequency is the ratio of a data value to the total of a row, a column, or the entire data set. It is expressed as a percent.<\/p>\n

Practice<\/strong><\/p>\n

The two-way table shows the eye color of 200 cats participating in a cat show.
\n\"Envision<\/p>\n

Question 1.
\nMake a two-way relative frequency table to show the distribution of the data with respect to gender. Round to the nearest tenth of a percent, as needed.
\nAnswer:
\nIt is given that
\nThe two-way table shows the eye color of 200 cats participating in a cat show.
\nNow,
\nThe given two-way frequency table is:
\n\"Envision
\nHence,
\nThe representation of the two-way relative frequency table that shows the distribution of the data wrt gender is:
\n\"\"<\/p>\n

Question 2.
\nWhat percent of cats that are female have blue eyes?
\nAnswer:
\nFrom the above two-way relative frequency table,
\nWe can observe that there are 30% of cats that are females who have blue eyes
\nHence, from the above,
\nWe can conclude that the percent of cats that are females and have blue eyes is: 30%<\/p>\n

Topic 4 Fluency Practice<\/h3>\n

Hidden Clue<\/strong>
\nFor each ordered pair, solve the equation to find the unknown coordinate. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help you answer the riddle below.<\/p>\n

\"Envision
\n\"Envision<\/p>\n

A (6, -0.5y + 20 – 0.5y = 13). 6, \"Envision<\/p>\n

B (4 – 3x \u2013 7x = -8, 7) \"Envision, 7<\/p>\n

C (2x + 4 – 6x = 24, 5) \"Envision, 5<\/p>\n

D (5x + 6 \u2013 10x = 31, 1) \"Envision, 1<\/p>\n

E (7x \u2013 3 \u2013 3x = 13, -2) \"Envision, -2<\/p>\n

F (4, -12y + 8y \u2013 21 = -5) 4, \"Envision<\/p>\n

G (44 = 6x \u2013 1 + 9x, \u20135) 4, \"Envision, -5<\/p>\n

H(-5, 4y + 14 \u2013 2y = 4) -5, \"Envision<\/p>\n

I (-5, 15+ y + 6 + 2y = 0) -5, \"Envision<\/p>\n

J (4, 3y + 32 – y = 18) 4, \"Envision<\/p>\n

K (6, 5y + 20 + 3y = -20) 6, \"Envision<\/p>\n

L (9x \u2013 14 \u2013 8x = -8, -1) \"Envision, -1<\/p>\n

M(-3, -5y + 10 – y = -2) -3, \"Envision<\/p>\n

N(-13 + x \u2013 5 \u2013 4x = -9, 4)
\nAnswer:
\nThe solutions of the above equations are:
\n\"\"<\/p>\n","protected":false},"excerpt":{"rendered":"

Practice with the help of enVision Math Common Core Grade 8 Answer Key Topic 4 Investigate Bivariate Data regularly and improve your accuracy in solving questions. enVision Math Common Core 8th Grade Answers Key Topic 4 Investigate Bivariate Data 3-ACT MATH Reach Out Reach for the skies! Who in your class can reach the highest? … Read more<\/a><\/p>\n","protected":false},"author":13,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[10],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/90191"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/13"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=90191"}],"version-history":[{"count":2,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/90191\/revisions"}],"predecessor-version":[{"id":90193,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/90191\/revisions\/90193"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=90191"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=90191"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=90191"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}