Exercises 1\u20132: What Is Random?<\/p>\n
Exercise 1.
\nWrite down a sequence of heads\/tails you think would typically occur if you tossed a coin 20 times. Compare your sequence to the ones written by some of your classmates. How are they alike? How are they different?
\nAnswer:
\nStudents might notice a lot of variability in the sequences. Most students will have at the most two heads or two tails in a row, and few will have more than one streak of three or more. Responses might look like the following:
\nH T H T H T H T H T H T H T H T H T H T
\nT T H T H T T H T T T H H T H T H H T T<\/p>\n
H T H H T H H T H H T H H T H H T H H T
\nH H T H T T H T H T T H T H T T H T H H<\/p>\n
Exercise 2.
\nWorking with a partner, toss a coin 20 times, and write down the sequence of heads and tails you get.
\na. Compare your results with your classmates\u2019.
\nb. How are your results from actually tossing the coin different from the sequences you and your classmates wrote down?
\nc. Toni claimed she could make up a set of numbers that would be random. What would you say to her?
\nAnswer:
\na. Students should notice that streaks of three or more heads and tails typically appear twice in the 20 tosses.
\nT T H T H H H H T T T T H T H H H T H T
\nH T H H H H H T T H T T H H T T T H T T<\/p>\n
T H H T T T T H H T H T H T T T H T T H
\nH T T T H H T H T H T T T T H T H T H H<\/p>\n
b. The results are different because when we generated the sequence, we did not have a lot of heads or tails in a row.
\nc. She could try, but she would probably not have some of the characteristics that a real set of random numbers would have, such as three consecutive numbers or four even numbers in a row.<\/p>\n
Exercises 3\u201311: Length of Words in the Poem \u201cCasey at the Bat\u201d<\/p>\n
Exercise 3.
\nSuppose you wanted to learn about the lengths of the words in the poem \u201cCasey at the Bat.\u201d You plan to select a sample of eight words from the poem and use these words to answer the following statistical question: On average, how long is a word in the poem? What is the population of interest here?
\nAnswer:
\nThe population of interest is all of the words in the poem.<\/p>\n
Exercise 4.
\nLook at the poem \u201cCasey at the Bat\u201d by Ernest Thayer, and select eight words you think are representative of words in the poem. Record the number of letters in each word you selected. Find the mean number of letters in the words you chose.
\nAnswer:
\nAnswers will vary. Sample response: The words their, while, thousand, ball, bat, strike, muscles, and grow would have a mean of 5.25 letters.<\/p>\n
Exercise 5.
\nA random sample is a sample in which every possible sample of the same size has an equal chance of being chosen. Do you think the set of words you wrote down was random? Why or why not?
\nAnswer:
\nAnswers will vary. Sample response: I thought it was random because I tried to use some little words and some long ones.<\/p>\n
Exercise 6.
\nWorking with a partner, follow your teacher\u2019s instructions for randomly choosing eight words. Begin with the title of the poem, and count a hyphenated word as one word.
\na. Record the eight words you randomly selected, and find the mean number of letters in those words.
\nb. Compare the mean of your random sample to the mean you found in Exercise 4. Explain how you found the mean for each sample.
\nAnswer:
\na. Sample response: We drew group 1, 12 (nine); group 1, 18 (four); group 5, 7 (seemed); group 3, 17 (in); group 23, 6 (fraud); group 27, 11 (is); group 27, 10 (air); group 16, 16 (close). The mean length of the words was 3.875.<\/p>\n
b. Sample response: The mean of the sample from Exercise 4 is based on the length of eight words I selected. The mean of the sample in this exercise is the mean of eight words randomly selected using the method of drawing numbers to represent the group number and word number. Anticipate that for most students, the mean from the random sample is lower than the mean for the self-selected sample.<\/p>\n
Exercise 7.
\nAs a class, compare the means from Exercise 4 and the means from Exercise 6. Your teacher will provide a chart to compare the means. Record your mean from Exercise 4 and your mean for Exercise 6 on this chart.
\n![\"Engage](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/03\/Engage-NY-Math-Grade-7-Module-5-Lesson-14-Exercise-Answer-Key-1.png\")
\nAnswer:
\nOrganize the responses in a table posted in the front of the class. Have students add their means to the poster. Consider the following example:
\n![\"Engage](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/03\/Engage-NY-Math-Grade-7-Module-5-Lesson-14-Exercise-Answer-Key-2.png\")
<\/p>\n
Exercise 8.
\nDo you think the means from Exercise 4 or the means from Exercise 6 are more representative of the mean of all of the words in the poem? Explain your choice.
\nAnswer:
\nSample response: The means in the random sample seem to be similar. As a result, I think the means from the random sample are more representative of the words in the poem.<\/p>\n
Exercise 9.
\nThe actual mean of the words in the poem \u201cCasey at the Bat\u201d is 4.2 letters. Based on the fact that the population mean is 4.2 letters, are the means from Exercise 4 or means from Exercise 6 a better representation of the mean of the population? Explain your answer.
\nAnswer:
\nSample response: The means from the random samples are similar and are closer to the mean of 4.2. Also, the means from Exercise 4 are generally larger than the mean of the population.<\/p>\n
Exercise 10.
\nHow did the population mean of 4.2 letters compare to the mean of your random sample from Exercise 6 and to the mean you found in Exercise 4?
\nAnswer:
\nSample response: The mean number of letters in all of the words in the poem is 4.2, which is about four letters per word, and the mean of my random sample, 3.875, was also about four letters per word. The mean of my sample in Exercise 4 was about five letters per word.<\/p>\n
Exercise 11.
\nSummarize how you would estimate the mean number of letters in the words of another poem based on what you learned in the above exercises.
\nAnswer:
\nSample response: Students should summarize a process similar to what they did in this lesson. They may simply indicate that they would number each word in the poem. They would make slips of paper from 1 to the number of words in the poem, place the slips of paper in a bag or jar, and select a sample of eight or more slips of paper. Students would then record the number of letters in the words identified by the slips of paper. As in the exercises, the mean of the sample would be used to estimate the mean of all of the words in the poem.<\/p>\n
Question 1.
\nWould any of the following provide a random sample of letters used in the text of the book Harry Potter and the Sorcerer\u2019s Stone by J.K. Rowling? Explain your reasoning.
\na. Use the first letter of every word of a randomly chosen paragraph.
\nb. Number all of the letters in the words in a paragraph of the book, cut out the numbers, and put them in a bag. Then, choose a random set of numbers from the bag to identify which letters you will use.
\nc. Have a family member or friend write down a list of his favorite words, and count the number of times each of the letters occurs.
\nAnswer:
\na. This is not a random sample. Some common letters, like u, do not appear very often as the first letter of a word and may tend to be underrepresented in the sample.
\nb. This would give you a random sample of the letters.
\nc. This would not be a random sample. He might like words that rhyme or that all start with the same letter. The list might also include words not in the book.<\/p>\n
Question 2.
\nIndicate whether the following are random samples from the given population, and explain why or why not.
\na. Population: All students in school; the sample includes every fifth student in the hall outside of class.
\nb. Population: Students in your class; the sample consists of students who have the letter s in their last names.
\nc. Population: Students in your class; the sample is selected by putting their names in a hat and drawing the sample from the hat.
\nd. Population: People in your neighborhood; the sample includes those outside in the neighborhood at 6:00 p.m.
\ne. Population: Everyone in a room; the sample is selected by having everyone toss a coin, and those that result in heads are the sample.
\nAnswer:
\na. Sample response: No. Not everyone in school would be in our hall before class. Our hall only has sixth graders in it, so the seventh and eighth graders would not have a chance to be chosen.
\nb. Sample response: No. Students who do not have the letter s in their last names would not have a chance to be chosen.
\nc. Sample response: Yes. Everyone would have the same chance to be chosen.
\nd. Sample response: No. People who are not in the neighborhood at that time have no chance of being selected.
\ne. Sample response: Yes. Everyone would have the same chance to be chosen.<\/p>\n
Question 3.
\nConsider the two sample distributions of the number of letters in randomly selected words shown below:
\n![\"Eureka](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/03\/Eureka-Math-7th-Grade-Module-5-Lesson-14-Problem-Set-Answer-Key-1.png\")
\na. Describe each distribution using statistical terms as much as possible.
\nb. Do you think the two samples came from the same poem? Why or why not?
\nAnswer:
\na. Answers will vary; the top distribution seems to have both a median and balance point, or mean, at 3, with a minimum of 1 letter in a word and a maximum of 7 letters. Most of the words in the sample were 2 to 4 letters long. The bottom distribution seems more skewed with the median of about 4 letters. The smallest number of letters was 2, and the largest was 10 letters. Most of the letters in this sample had between 2 and 5 letters.<\/p>\n
b. Sample response: The samples could have come from the same poem, but the distributions seem different both with respect to shape and to measure of center, so it seems more likely that they were from two different populations.<\/p>\n
Question 4.
\nWhat questions about samples and populations might you want to ask if you saw the following headlines in a newspaper?
\na. \u201cPeach Pop is the top flavor according to 8 out of 10 people.\u201d
\nb. \u201cCandidate X looks like a winner! 10 out of 12 people indicate they will vote for Candidate X.\u201d
\nc. \u201cStudents overworked. Over half of 400 people surveyed think students spend too many hours on homework.\u201d
\nd. \u201cAction\/adventure was selected as the favorite movie type by an overwhelming 75% of those surveyed.\u201d
\nAnswer:
\na. Sample response: How were the people selected? How many people were surveyed? What were the choices, and how many did not like Peach Pop?
\nb. Sample response: How was the sample chosen? Were the people selected at random, or were they friends of Candidate X?
\nc. Sample response: Who was surveyed, and how were they selected? Was the survey given to students in a school?
\nd. Sample response: Was the survey given at a movie theater showing an action\/adventure movie where people were there because they like that kind of movie?<\/p>\n
Question 1.
\nWrite down three things you learned about taking a sample from the work we have done today.
\nAnswer:
\nA random sample is one where every element in the set has an equal chance of being selected.
\nWhen people just choose a sample they think will be random, it will usually be different from a real random sample.
\nRandom samples are usually similar to the population.<\/p>\n","protected":false},"excerpt":{"rendered":"
Engage NY Eureka Math 7th Grade Module 5 Lesson 14 Answer Key Eureka Math Grade 7 Module 5 Lesson 14 Exercise Answer Key Exercises 1\u20132: What Is Random? Exercise 1. Write down a sequence of heads\/tails you think would typically occur if you tossed a coin 20 times. Compare your sequence to the ones written … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[6],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/75243"}],"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\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=75243"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/75243\/revisions"}],"predecessor-version":[{"id":92164,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/75243\/revisions\/92164"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=75243"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=75243"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=75243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}