{"id":89683,"date":"2022-05-23T10:15:26","date_gmt":"2022-05-23T04:45:26","guid":{"rendered":"https:\/\/ccssanswers.com\/?p=89683"},"modified":"2022-05-10T14:20:55","modified_gmt":"2022-05-10T08:50:55","slug":"eureka-math-algebra-2-module-4-lesson-4","status":"publish","type":"post","link":"https:\/\/ccssanswers.com\/eureka-math-algebra-2-module-4-lesson-4\/","title":{"rendered":"Eureka Math Algebra 2 Module 4 Lesson 4 Answer Key"},"content":{"rendered":"

Engage NY Eureka Math Algebra 2 Module 4 Lesson 4 Answer Key<\/h2>\n

Eureka Math Algebra 2 Module 4 Lesson 4 Example Answer Key<\/h3>\n

Exercise 1:<\/p>\n

In previous lessons, conditional probabilities were used to investigate whether or not there is a connection between two events. This lesson formalizes this idea and introduces the concept of independence.<\/p>\n

1.
\nSeveral questions are posed below. Each question is about a possible connection between two events. For each question, identify the two events, and indicate whether or not you think that there would be a connection. Explain your reasoning.<\/p>\n

a. Are high school students whose parents or guardians set a midnight curfew less likely to have a traffic violation
\nthan students whose parents or guardians have not set such a curfew?
\nAnswer:
\nResponses vary.
\nThe two events are parents set a midnight curfew\u009d and students have o traffic violation.\u009d Anticipate that students may indicate either that students with a curfew are less likely to have a traffic violation or that there is no connection. Either answer is acceptable as long as students provide an explanation.<\/p>\n

b. Are left-handed people more likely than right-handed people to be interested in the arts?
\nAnswer:
\nResponses vary.
\nThe two events are people are right-handed\u009d and people are interested in the arts. Students may argue that there is a connection or that there is nota connection between being right-handed and being interested in the arts. Look for some reasoning that indicates that if one of the descriptions is met (selecting a right- handed person), it is more (or less) likely this person is also interested in the arts. Students may indicate that they do not think that there is a connection, which is acceptable as long as they explain their reasoning.<\/p>\n

c. Are students who regularly listen to classical music more likely to be interested in mathematics than students who do not regularly listen to classical music?
\nAnswer:
\nResponses vary.
\nThe two events are \u00e2\u20ac\u0153students regularly listen to classical music\u00e2\u20ac\u009d and \u00e2\u20ac\u0153students are interested in mathematics. Students may argue that there is a connection or there is not a connection between classical music and an interest in mathematics. Look for some reasoning that indicates that the fane of the descriptions is met, it is more likely the other description will or will not occur.<\/p>\n

d. Are people who play video games more than 10 hours per week more likely to select football as their favorite sport than people who do not play video games more than 10 hours per week?
\nAnswer:
\nResponses vary.
\nThe two events are people who play video games more than 10 hours per week and people whose favorite sport is football.\u009d Students may argue that there is a connection or there is not a connection between video games and an interest in football, which is acceptable as long as they explain their reasoning.<\/p>\n

Two events are independent when knowing that one event has occurred does not change the likelihood that the second event has occurred. How can conditional probabilities be used to tell if two events are independent or not independent?<\/p>\n

Exercises 2 – 6:<\/p>\n

Recall the hypothetical 1000 two-way frequency table that was used to classify students at Rufus King High School according to gender and whether or not they participated in an after-school athletics program.<\/p>\n

Table 1: Participation of Male and Female Students in an After-School Athletics Program<\/p>\n\n\n\n\n\n\n
<\/td>\nParticipate in an After-School\u00a0 Athletics Program<\/td>\nDo Not Participate in an
\nAfter-School Athletics Program<\/td>\n		Total<\/td>\n<\/tr>\n
Female<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
Male<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
Total<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Answer:<\/p>\n\n\n\n\n\n\n
<\/td>\nParticipate in an After-School\u00a0 Athletics Program<\/td>\nDo Not Participate in an
\nAfter-School Athletics Program<\/td>\n		Total<\/td>\n<\/tr>\n
Female<\/td>\n232<\/td>\n348<\/td>\n580<\/td>\n<\/tr>\n
Male<\/td>\n168<\/td>\n252<\/td>\n420<\/td>\n<\/tr>\n
Total<\/td>\n400<\/td>\n600<\/td>\n1,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Exercise 2.
\nFor each of the following, indicate whether the probability described is one that can be calculated using the values in Table 1. Also indicate whether or not it is a conditional probability.<\/p>\n

a. The probability that a randomly selected student participates in an after-school athletics program
\nAnswer:
\nThis probability can be calculated from the values in the table. It is not a conditional probability. The probability is based on the entire school population.<\/p>\n

b. The probability that a randomly selected student who is female participates in an after-school athletics program
\nAnswer:
\nThis probability can be calculated from the values in the table. it is a conditional probability because it is a probability-based on only the female students at the school.<\/p>\n

c. The probability that a randomly selected student who is male participates in an after-school athletics program
\nAnswer:
\nThis probability can be calculated from the values in the table. It is a conditional probability because it is based on only the male students at the school.<\/p>\n

Exercise 3.
\nUse Table 1 to calculate each of the probabilities described in Exercise 2.<\/p>\n

a. The probability that a randomly selected student participates in an after-school athletics program
\nAnswer:
\n\\(\\frac{400}{1000}\\) = 0. 400 (This means that 40% of all students participate in after-school athletics programs.)<\/p>\n

b. The probability that a randomly selected student who is female participates in an after-school athletics
\nprogram
\nAnswer:
\n\\(\\frac{232}{580}\\) = 0.400 (This means that 40% of female students participate in after-school athletics programs.)<\/p>\n

c. The probability that a randomly selected student who is male participates in an after-school athletics program
\nAnswer:
\n\\(\\frac{168}{420}\\) = 0.400 (This means that 40% of male students participate in after-school athletics programs.)<\/p>\n

Exercise 4.
\nWould your prediction of whether or not a student participates in an after-school athletics program change if you knew the gender of the student? Explain your answer.
\nAnswer:
\nNo. Based on the conditional probabilities, the probability that o student participates in an after-school athletics program is 0.400 for both male and female students.<\/p>\n

Two events are independent if knowing that one event has occurred does not change the probability that the other event has occurred. For example, consider the following two events:<\/p>\n

F: The event that a randomly selected student is female
\nS: The event that a randomly selected student participates in an after-school athletics program.<\/p>\n

Events F and S would be independent if the probability that a randomly selected student participates in an after-school athletics program is equal to the probability that a randomly selected student who is female participates in an after-school athletics program.<\/p>\n

If this was the case, knowing that a randomly selected student female does not change the probability that the selected student participates in an after-school athletics program. Then, F and S would be independent.<\/p>\n

Exercise 5.
\nBased on the definition of independence, are the events randomly selected student is female\u009d and randomly selected student participates in an after-school athletics program\u009d independent? Explain.
\nAnswer:
\nYes, they are independent because knowing that a randomly selected student is female does not change the probability that the selected student participates in an after-school athletics program. The probability that a randomly selected student participates in an after-school athletics program is 0.400, and the probability that a randomly selected student who is female participates is also 0.400.<\/p>\n

Exercise 6.
\nA randomly selected student participates in an after-school athletics program.<\/p>\n

a. What is the probability this student is female?
\nAnswer:
\n\\(\\frac{232}{400}\\) = 0.58
\nThis is equal to the probability that a randomly selected student is female.<\/p>\n

b. Using only your answer from part (a), what is the probability that this student is male? Explain how you arrived at your answer.
\nAnswer:
\nThe probability is 0.42.
\nThe probability in part (a) can be interpreted as 58% of the students who participate in after-school athletics programs are female. The rest must be male, so the probability that a randomly selected student who participates in an after-school athletics program is male is 0.42. 1 – 0.58 = 0.42<\/p>\n

Exercises 7 – 11:<\/p>\n

Consider the data below.<\/p>\n\n\n\n\n\n\n
<\/td>\nNo Household Member Smokes<\/td>\nAt Least One Household Member Smokes<\/td>\nTotal<\/td>\n<\/tr>\n
Student Has Asthma<\/td>\n69<\/td>\n113<\/td>\n182<\/td>\n<\/tr>\n
Student Does Not Have Asthma<\/td>\n473<\/td>\n282<\/td>\n755<\/td>\n<\/tr>\n
Total<\/td>\n542<\/td>\n395<\/td>\n937<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Exercise 7.
\nYou are asked to determine if the two events a randomly selected student has asthma\u009d and a randomly selected student has a household member who smokes\u009d are independent. What probabilities could you calculate to answer this question?
\nAnswer:
\nStudents could indicate that the probability of selecting a student who has a household member who smokes is the same for students who have asthma as for those who do not have asthma. Or students could indicate that the probability of selecting a student who has a household member who smokes from the students who have asthma would need to be equal to the probability of selecting a student who has a household member who smokes from all of the students.<\/p>\n

Or students could Indicate that the probability of selecting a student who has a household member who smokes from the students who do not have asthma would need to be equal to the probability of selecting a student who has a household member who smokes from all of the students.<\/p>\n

Students could also indicate that conditional probabilities based on the columns would have to be equal for the events to be independent. The probability of selecting a student who has asthma from the students who have no household member who smokes would need to be equal to the probability of selecting a student who has asthma from the students who have at least one household member who smokes.<\/p>\n

Exercise 8.
\nCalculate the probabilities you described in Exercise 7.
\nAnswer:
\nThe row conditional probabilities described are as follows:
\n\\(\\frac{69}{182}\\) = 0.379<\/p>\n

\\(\\frac{473}{755}\\) = 0.626<\/p>\n

The column conditional probabilities described ore as follows:
\n\\(\\frac{69}{542}\\) = 0.127<\/p>\n

\\(\\frac{113}{395}\\) = 0.286<\/p>\n

Exercise 9.
\nBased on the probabilities you calculated in Exercise 8, are these two events independent or not independent? Explain.
\nAnswer:
\nNo. The conditional probabilities need to be equal for the events to be independent.<\/p>\n

Exercise 10.
\nIs the probability that a randomly selected student who has asthma and who has a household member who smokes the same as or different from the probability that a randomly selected student who does not have asthma but does have a household member who smokes? Explain your answer.
\nAnswer:
\nThe probabilities are different as the events are not independent. The probability of a randomly selected student
\nwho has asthma having a household member who smokes is \\(\\frac{113}{182^{\\prime}}\\) which is approximately 0.62. The probability of a randomly selected student who does not have asthma having a household member who smokes is \\(\\frac{282}{755^{\\prime}}\\) which is approximately 0.37.<\/p>\n

Exercise 11.
\nA student is selected at random. The selected student indicates that he has a household member who smokes. What is the probability that the selected student has asthma?
\nAnswer:
\n\\(\\frac{113}{395}\\) = 0.29 This is the conditional probability that a randomly selected student who has a household family member who smokes has asthma.<\/p>\n

Eureka Math Algebra 2 Module 4 Lesson 4 Problem Set Answer Key<\/h3>\n

Question 1.
\nConsider the following questions:<\/p>\n

a. A survey of the students at a Midwest high school asked the following questions:
\nDo you use a computer at least 3 times a week to complete your schoolwork?
\nAre you taking a mathematics class?
\nDo you think the events “a randomly selected student uses a computer at least 3 times a week”\u009d and “a randomly selected student is taking a mathematics class\u00e2\u20ac\u009d are independent or not independent? Explain your reasoning.
\nAnswer:
\nAnticipate students indicate that using a computer at least 3 times per week and taking a mathematics class are not independent. However, it is also acceptable for students to make a case for independence. Examine the explanations students provide to see if they understand the meaning of independence.<\/p>\n

b. The same survey also asked students the following:
\n\u201cDo you participate in any extracurricular activities at your school?\u201d
\n\u201cDo you know what you want to do after high school?\u201d
\nDo you think the events \u201ca randomly selected student participates in extracurricular activities\u201d and \u201ca randomly selected student knows what she wants to do after completing high school\u201d are independent or not, independent? Explain your reasoning.
\nAnswer:
\nAnswers will vary. Anticipate students indicate that students involved in extracurricular activities are often students who want to attend college. It is likely the events are not independent.<\/p>\n

c. People attending a professional football game in 2013 completed a survey that included the following questions:
\n\u201cIs this the first time you have attended a professional football game?\u201d
\n\u201cDo you think football is too violent?\u201d
\nDo you think the events \u201ca randomly selected person who completed the survey Is attending a professional football game for the first time\u201d and \u201ca randomly selected person who completed the survey thinks football is too violent\u201d are independent or not independent? Explain your reasoning.
\nAnswer:
\nAnswers will vary. Anticipate that students indicate that people who attend football more often are more likely to not think the game is too violent it is likely the events are not independent. Again, examine the explanations students provide if they indicate the events are not independent.<\/p>\n

Question 2.
\nComplete the table below in a way that would indicate the two events \u201cuses a computer\u201d and \u201cis taking a mathematics class\u201d are independent.<\/p>\n\n\n\n\n\n\n
<\/td>\nUses a Computer at Least \u00a0Times a Week for Schoolwork<\/td>\nDoes Not Use a Computer at Least \u00a0Times a Week for Schoolwork<\/td>\nTotal<\/td>\n<\/tr>\n
In a Mathematics Class<\/td>\n<\/td>\n<\/td>\n700<\/td>\n<\/tr>\n
Not in a Mathematics Class<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
Total<\/td>\n600<\/td>\n<\/td>\n1,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Answer:<\/p>\n\n\n\n\n\n\n
<\/td>\nUses a Computer at Least \u00a0Times a Week for Schoolwork<\/td>\nDoes Not Use a Computer at Least \u00a0Times a Week for Schoolwork<\/td>\nTotal<\/td>\n<\/tr>\n
In a Mathematics Class<\/td>\n420<\/td>\n280<\/td>\n700<\/td>\n<\/tr>\n
Not in a Mathematics Class<\/td>\n180<\/td>\n120<\/td>\n300<\/td>\n<\/tr>\n
Total<\/td>\n600<\/td>\n400<\/td>\n1,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Question 3.
\nComplete the following hypothetical 1000 table. Are the events \u201cparticipates in extracurricular activities\u201d and \u201cknow what I want to do after high school\u201d independent or not independent? Justify your answer.<\/p>\n\n\n\n\n\n\n
<\/td>\nParticipates in Extracurricular Activities<\/td>\nDoes Not Participate in Extracurricular Activities<\/td>\nTotal<\/td>\n<\/tr>\n
Know What I Want to Do After High School<\/td>\n<\/td>\n<\/td>\n800<\/td>\n<\/tr>\n
Do Not Know What I Want to Do After High School<\/td>\n50<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
Total<\/td>\n600<\/td>\n<\/td>\n1,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Answer:<\/p>\n\n\n\n\n\n\n
<\/td>\nParticipates in Extracurricular Activities<\/td>\nDoes Not Participate in Extracurricular Activities<\/td>\nTotal<\/td>\n<\/tr>\n
Know What I Want to Do After High School<\/td>\n550<\/td>\n250<\/td>\n800<\/td>\n<\/tr>\n
Do Not Know What I Want to Do After High School<\/td>\n50<\/td>\n150<\/td>\n200<\/td>\n<\/tr>\n
Total<\/td>\n600<\/td>\n400<\/td>\n1,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The events \u201cstudent participates in extracurricular activities\u201d and \u201cstudent knows what I want to do after high school are not independent. Students could indicate that the events are not independent in several ways.<\/p>\n

For example, \\(\\frac{50}{200}\\) (the probability that a randomly selected student who does not know what he wants to do after high school participates in extracurricular activities) does not equal \\(\\frac{550}{800}\\) (the probability that a randomly selected student who does know what he wants to do after high school participates in extracurricular activities).<\/p>\n

Question 4.
\nThe following hypothetical 1000 table is from lesson 2.<\/p>\n\n\n\n\n\n\n
<\/td>\nNo Household Member Smokes<\/td>\nAt Least One Household Member Smokes<\/td>\nTotal<\/td>\n<\/tr>\n
Student Has Asthma<\/td>\n73<\/td>\n120<\/td>\n193<\/td>\n<\/tr>\n
Student Does Not Have Asthma<\/td>\n506<\/td>\n301<\/td>\n807<\/td>\n<\/tr>\n
Total<\/td>\n579<\/td>\n421<\/td>\n1,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The actual data from the entire population are given in the table below.<\/p>\n\n\n\n\n\n\n
<\/td>\nNo Household Member Smokes<\/td>\nAt Least One Household Member Smokes<\/td>\nTotal<\/td>\n<\/tr>\n
Student Has Asthma<\/td>\n69<\/td>\n113<\/td>\n182<\/td>\n<\/tr>\n
Student Does Not Have Asthma<\/td>\n473<\/td>\n282<\/td>\n755<\/td>\n<\/tr>\n
Total<\/td>\n542<\/td>\n395<\/td>\n937<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

a. Based on the hypothetical 1000 table, what is the probability that a randomly selected student who has asthma has at least one household member who smokes?
\nAnswer:
\n\\(\\frac{120}{193}\\) = 0.622<\/p>\n

b. Based on the actual data, what is the probability that a randomly selected student who has asthma has at least one household member who smokes (round your answer to 3 decimal places)?
\nAnswer:
\n\\(\\frac{113}{182}\\) = 0.621<\/p>\n

c. Based on the hypothetical 1000 table, what is the probability that a randomly selected student who has no household member who smokes has asthma?
\nAnswer:
\n\\(\\frac{73}{579}\\) = 0.126<\/p>\n

d. Based on the actual data, what is the probability that a randomly selected student who has no household member who smokes has asthma?
\nAnswer:
\n\\(\\frac{69}{542}\\) = 0.127<\/p>\n

e. What do you notice about the probabilities calculated from the actual data and the probabilities calculated
\nfrom the hypothetical 1000 table?
\nAnswer:
\nThe conditional probabilities differ only due to rounding in constructing the hypothetical 1000 table from probability information based on the actual data. When an actual data table is available, it can be used to calculate probabilities.<\/p>\n

When only probability information is available, constructing a hypothetical 1000 table from that information and using it to compute other probabilities will give some answers as if the actual data were available.<\/p>\n

Question 5.
\nAs part of the asthma research, the investigators wondered if students who have asthma are less likely to have a pet at home than students who do not have asthma. They asked the following two questions:
\n\u201cDo you have asthma?”
\n“Do you have a pet at homer?”
\nBased on the responses to these questions, you would like to set up a two-way table that you could use to determine if the following two events are independent or not independent:
\nEvent 1: A randomly selected student has asthma.
\nEvent 2: A randomly selected student has a pet at home.<\/p>\n

a. How would you label the rows of the two-way table?
\nAnswer:
\nAnticipate students indicate for the rows \u201cHas Asthma\u201d and \u201cDoes Not Have Asthma.\u201d Students might use these labels for the columns rather thon the rows, which is also acceptable.<\/p>\n

b. How would you label the columns of the two-way table?
\nAnswer:
\nAnticipate students indicate for the columns \u201cHas a Pet\u201d and \u201cDoes Not Have a Pet.\u201d<\/p>\n

c. What probabilities would you calculate to determine if Event 1 and Event 2 are independent?
\nAnswer:
\nAnswers may vary. Row conditional probabilities or column conditional probabilities would have to be equal if the events are independent. For column conditional probabilities (based on the definition of rows and columns above), this would mean that the probability that a randomly selected student who has a pet has asthma is equal to the probability that a randomly selected student who does not have a pet has asthma.<\/p>\n

Eureka Math Algebra 2 Module 4 Lesson 4 Exit Ticket Answer Key<\/h3>\n

Question 1.
\nThe following hypothetical 1000 two-way table was introduced in the previous lesson:
\n\"Eureka<\/p>\n

Are the events \u201ca randomly selected person watched the online ad\u201d and \u201ca randomly selected person plans to vacation in New York within the next year\u2019 independent or not independent? Justify your answer using probabilities calculated from the information in the table.
\nAnswer:
\n\"Eureka<\/p>\n

The conditional probabilities that could be used to evaluate if the events are independent are the probability that given the selected member watched the online ad, the member plans to vacation in New York, and the probability that given the selected member did not watch the ad, the member plans to vacation in New York. Because the probabilities are equal, the events are independent.
\n\\(\\frac{300}{750}\\) = 0.40 and \\(\\frac{100}{250}\\) = 0.40<\/p>\n

Question 2.
\nA survey conducted at a local high school indicated that 30% of students have a job during the school year. If having a job and being in the eleventh grade are not independent, what do you know about the probability that a randomly selected student who is in the eleventh grade would have a job? Justify your answer.
\nAnswer:
\nThe probability that a student selected from the eleventh grade has a job would not be equal to 0.30. Not independent means that knowing the selected student is in the eleventh grade changes the probability that the student has a job.<\/p>\n

Question 3.
\nEighty percent of the dogs at a local kennel are in good health. If the events \u201ca randomly selected dog at this kennel is in good health\u201d and \u201ca randomly selected dog at this kennel weighs more than 30 pounds\u201d are independent, what do you know about the probability that a randomly selected dog that weighs more than 30 pounds will be in good health? Justify your answer.
\nAnswer:
\nAs the events are independent, knowing that the selected dog weighs more than 30 pounds does not change the probability that the dog is in good health. This means that the probability that a large dog is in good health is also 0.80.<\/p>\n","protected":false},"excerpt":{"rendered":"

Engage NY Eureka Math Algebra 2 Module 4 Lesson 4 Answer Key Eureka Math Algebra 2 Module 4 Lesson 4 Example Answer Key Exercise 1: In previous lessons, conditional probabilities were used to investigate whether or not there is a connection between two events. This lesson formalizes this idea and introduces the concept of independence. … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/89683"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=89683"}],"version-history":[{"count":1,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/89683\/revisions"}],"predecessor-version":[{"id":136829,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/89683\/revisions\/136829"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=89683"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=89683"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=89683"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}