Deck 9: Comparing More Than Two Proportions

Question 1

Researchers analyzed eating behavior and obesity at Chinese buffets. They estimated people's body mass indexes (BMI) as they entered the restaurant then categorized them into three groups---bottom third (lightest), middle third, and top third (heaviest). One variable they looked at was whether or not they browsed the buffet (looked it over) before serving themselves or served themselves immediately. Treating the BMI categories as the explanatory variable and whether or not they browsed first as the response, the researchers wanted to see if there was an association between BMI and whether or not they browsed the buffet before serving themselves. They found the following results:
• Bottom Third: 35 of the 50 people browsed first
• Middle Third: 24 of the 50 people browsed first
• Top Third: 17 of the 50 people browsed first
-Fill in the following two-way table with the observed data.

Answer 1

Tol for each value = +/- 0

Question 2

Researchers analyzed eating behavior and obesity at Chinese buffets. They estimated people's body mass indexes (BMI) as they entered the restaurant then categorized them into three groups---bottom third (lightest), middle third, and top third (heaviest). One variable they looked at was whether or not they browsed the buffet (looked it over) before serving themselves or served themselves immediately. Treating the BMI categories as the explanatory variable and whether or not they browsed first as the response, the researchers wanted to see if there was an association between BMI and whether or not they browsed the buffet before serving themselves. They found the following results:
• Bottom Third: 35 of the 50 people browsed first
• Middle Third: 24 of the 50 people browsed first
• Top Third: 17 of the 50 people browsed first
-For each BMI category, calculate the conditional proportion of people that browsed first and write these numbers below as decimals, not fractions.
Bottom Third___________ Middle Third____________ Top Third____________

Answer 2

Bottom Third = 35/50 = 0.7; Middle Third = 0.48, Top Third = 0.34

Question 3

Researchers analyzed eating behavior and obesity at Chinese buffets. They estimated people's body mass indexes (BMI) as they entered the restaurant then categorized them into three groups---bottom third (lightest), middle third, and top third (heaviest). One variable they looked at was whether or not they browsed the buffet (looked it over) before serving themselves or served themselves immediately. Treating the BMI categories as the explanatory variable and whether or not they browsed first as the response, the researchers wanted to see if there was an association between BMI and whether or not they browsed the buffet before serving themselves. They found the following results:
• Bottom Third: 35 of the 50 people browsed first
• Middle Third: 24 of the 50 people browsed first
• Top Third: 17 of the 50 people browsed first

-What is the null hypothesis to determine if there is an association between BMI and if a person browses first?

A) H _₀:?₁=?₂=?₃=0
B) H_₀:x ?₁=x ?₂=x ?₃=0
C) H_₀:?₁=?₂=?₃=0
D) H_₀:p ?₁=p ?₂=p ?₃=0

Answer 3

H_₀:?₁=?₂=?₃=0

Question 4

Researchers analyzed eating behavior and obesity at Chinese buffets. They estimated people's body mass indexes (BMI) as they entered the restaurant then categorized them into three groups---bottom third (lightest), middle third, and top third (heaviest). One variable they looked at was whether or not they browsed the buffet (looked it over) before serving themselves or served themselves immediately. Treating the BMI categories as the explanatory variable and whether or not they browsed first as the response, the researchers wanted to see if there was an association between BMI and whether or not they browsed the buffet before serving themselves. They found the following results:
• Bottom Third: 35 of the 50 people browsed first
• Middle Third: 24 of the 50 people browsed first
• Top Third: 17 of the 50 people browsed first
-Compute the Mean Group Diff statistic.

Question 5

Researchers analyzed eating behavior and obesity at Chinese buffets. They estimated people's body mass indexes (BMI) as they entered the restaurant then categorized them into three groups---bottom third (lightest), middle third, and top third (heaviest). One variable they looked at was whether or not they browsed the buffet (looked it over) before serving themselves or served themselves immediately. Treating the BMI categories as the explanatory variable and whether or not they browsed first as the response, the researchers wanted to see if there was an association between BMI and whether or not they browsed the buffet before serving themselves. They found the following results:
• Bottom Third: 35 of the 50 people browsed first
• Middle Third: 24 of the 50 people browsed first
• Top Third: 17 of the 50 people browsed first
-Use the Multiple Proportions applet to calculate an approximate p-value for this test. Use at least 1,000 shuffles.

Question 6

Researchers analyzed eating behavior and obesity at Chinese buffets. They estimated people's body mass indexes (BMI) as they entered the restaurant then categorized them into three groups---bottom third (lightest), middle third, and top third (heaviest). One variable they looked at was whether or not they browsed the buffet (looked it over) before serving themselves or served themselves immediately. Treating the BMI categories as the explanatory variable and whether or not they browsed first as the response, the researchers wanted to see if there was an association between BMI and whether or not they browsed the buffet before serving themselves. They found the following results:
• Bottom Third: 35 of the 50 people browsed first
• Middle Third: 24 of the 50 people browsed first
• Top Third: 17 of the 50 people browsed first

-What is the shape of the resulting null distribution of MAD statistics shown in the output of the Multiple Proportions applet?

A) Symmetric
B) Normal
C) Left skewed
D) Right skewed

Question 7

Researchers analyzed eating behavior and obesity at Chinese buffets. They estimated people's body mass indexes (BMI) as they entered the restaurant then categorized them into three groups---bottom third (lightest), middle third, and top third (heaviest). One variable they looked at was whether or not they browsed the buffet (looked it over) before serving themselves or served themselves immediately. Treating the BMI categories as the explanatory variable and whether or not they browsed first as the response, the researchers wanted to see if there was an association between BMI and whether or not they browsed the buffet before serving themselves. They found the following results:
• Bottom Third: 35 of the 50 people browsed first
• Middle Third: 24 of the 50 people browsed first
• Top Third: 17 of the 50 people browsed first

-Based upon the p-value of 0.001, what is the appropriate conclusion for this test?

A) We have strong evidence of an association between BMI and if a person browses first among all people who eat at Chinese buffets.
B) We have strong evidence of an association between BMI and if a person browses first among people who eat at Chinese buffets similar to those in the study.
C) We have strong evidence of no association between BMI and if a person browses first among all people who eat at Chinese buffets.
D) We have strong evidence of no association between BMI and if a person browses first among people who eat at Chinese buffets similar to those in the study.

Question 8

Which of the following is not one of the 3 "S"'s in the 3S strategy?

A) Simulate
B) Statistic
C) Sample
D) Strength of evidence

Question 9

Suppose three students were interested in studying the association between a person's political party (Democrat, Republican, Independent, or Other), and whether they voted in the last election (Yes, No). Each of the three students selects a random sample of 40 students and asks each student their political party and if they voted in the last election. Their three Mean Group Diff statistics are:
Mean Group Diff 1 = 0.12 Mean Group Diff 2 = 0.35 Mean Group Diff 3 = 0.27
Which Mean Group Diff statistic provides the strongest evidence against the null hypothesis?

A) 0.12
B) 0.35
C) 0.27

Question 10

As the Mean Group Diff statistic increases, the p-value decreases.

Question 11

The Mean Group Diff statistic can never be negative.

Question 12

The Mean Group Diff statistic is the average distance sample proportions are from each other.

Question 13

The Mean Group Diff statistic is the average of a group of sample proportions.

Question 14

Is there an association between a person's political affiliation (Democrat, Republican, or Inde-pendent) and whether or not they give to charity?

-Describe the appropriate parameter(s) and assign symbols(s) to the parameters. Select all that apply

A) ?_Dem = probability that a Democrat will give to charity
B) ?_Rep = probability that a Republican will give to charity
C) ?_Ind = probability that a Independent will give to charity

Question 15

Is there an association between a person's political affiliation (Democrat, Republican, or Inde-pendent) and whether or not they give to charity?

-Assume that, in reality, there is no association between a person's political affiliation and whether or not they give to charity. Which of the following has the largest probability?

A) The p-value for a test of H ₀:? _Dem=? _Rep is greater than 0.05.
B) The p-value for a test of H ₀:? _Dem=? _Rep is greater than 0.05 AND the p-value for a test of H ₀:? _Dem=? _Ind is greater than 0.05.
C) The p-value for a test of H ₀:? _Dem=? _Rep is greater than 0.05 AND the p-value for a test of H ₀:? _Dem=? Ind is greater than 0.05 AND the p-value for a test of H ₀:? _Ind=? _Rep is greater than 0.05.
D) All of the above have equal probability.

Question 16

Hope student researchers want to determine if students yawning (when being yawned at) is asso-ciated with the time of day (morning, afternoon, and evening). They found the following results when they yawned in front of people around campus:

• Morning: 25 of the 39 people yawned
• Afternoon: 15 of the 38 people yawned
• Evening: 29 of the 44 people yawned
-Fill in the following two-way table with the observed data.

Question 17

Hope student researchers want to determine if students yawning (when being yawned at) is asso-ciated with the time of day (morning, afternoon, and evening). They found the following results when they yawned in front of people around campus:

• Morning: 25 of the 39 people yawned
• Afternoon: 15 of the 38 people yawned
• Evening: 29 of the 44 people yawned

-Are validity conditions met to conduct a chi-square test of association?

A) Yes, since 39, 38, and 44 are all larger than 20.
B) Yes, since 121 is larger than 20.
C) Yes, since each cell in the table has at least 10 observations.
D) No, since several cells in the table have less than 20 observations.

Question 18

Hope student researchers want to determine if students yawning (when being yawned at) is asso-ciated with the time of day (morning, afternoon, and evening). They found the following results when they yawned in front of people around campus:

• Morning: 25 of the 39 people yawned
• Afternoon: 15 of the 38 people yawned
• Evening: 29 of the 44 people yawned
-For each time of day, calculate the conditional proportion of people that yawned and write these numbers as decimals not fractions.
Morning___________ Afternoon____________ Evening____________

Question 19

Hope student researchers want to determine if students yawning (when being yawned at) is asso-ciated with the time of day (morning, afternoon, and evening). They found the following results when they yawned in front of people around campus:

• Morning: 25 of the 39 people yawned
• Afternoon: 15 of the 38 people yawned
• Evening: 29 of the 44 people yawned

-Which applet would you use to conduct a theory-based test of association between time of day and whether a person yawns?

A) One Proportion
B) Two Proportions
C) Multiple Proportions
D) Theory-Based Inference

Question 20

Hope student researchers want to determine if students yawning (when being yawned at) is asso-ciated with the time of day (morning, afternoon, and evening). They found the following results when they yawned in front of people around campus:

• Morning: 25 of the 39 people yawned
• Afternoon: 15 of the 38 people yawned
• Evening: 29 of the 44 people yawned

-When this test is run, a p-value of 0.03 is found. What would be an appropriate conclusion?

A) We have strong evidence that the probability a person yawns is different between all three times of day: morning, afternoon, and evening.
B) We have strong evidence of an association between time of day and whether a person yawns.
C) We have strong evidence of no association between time of day and whether a person yawns.
D) We do not have strong evidence of an association between time of day and whether a person yawns.

Question 21

Hope student researchers want to determine if students yawning (when being yawned at) is asso-ciated with the time of day (morning, afternoon, and evening). They found the following results when they yawned in front of people around campus:

• Morning: 25 of the 39 people yawned
• Afternoon: 15 of the 38 people yawned
• Evening: 29 of the 44 people yawned

-The chi-square statistic for this test was 6.991. If instead, the chi-square statistic had been 5.332, would the p-value be smaller, larger, or remain the same?

A) Smaller
B) Larger
C) Remain the same

Question 22

Hope student researchers want to determine if students yawning (when being yawned at) is asso-ciated with the time of day (morning, afternoon, and evening). They found the following results when they yawned in front of people around campus:

• Morning: 25 of the 39 people yawned
• Afternoon: 15 of the 38 people yawned
• Evening: 29 of the 44 people yawned

-Follow-up confidence intervals for all pairwise differences in proportions are as follows:
•0.0299 to 0.4627: Morning - Afternoon
•-0.2237 to 0.1876: Morning - Evening
•-0.4736 to -0.0551: Afternoon - Evening
Based off of these intervals, which of the following conclusions are correct?

A) The probability of yawning in the morning differs significantly from the probability of yawning in the evening.
B) The probability of yawning in the morning differs significantly from the probability of yawning the afternoon.
C) The probability of yawning in the afternoon does not differ significantly from the probability of yawning in the evening.
D) The probability of yawning in the morning does not differ significantly from the probability of yawning in the afternoon.

Question 23

Can telling a joke affect whether or not a customer in a coffee bar leaves a tip for the waiter? A study investigated this question at a coffee bar in France. The waiter randomly assigned coffee-ordering customers into three groups: one received a card telling a joke with their bill, another group received a card containing an advertisement for a local restaurant, and the third received no card. The results are shown below.

-Are validity conditions met to conduct a chi-square test of association?

A) Yes, since 80 is larger than 20.
B) Yes, since 240 is larger than 20.
C) Yes, since each cell in the table has at least 10 observations.
D) No, we did not collect a random sample of customers.

Question 24

Can telling a joke affect whether or not a customer in a coffee bar leaves a tip for the waiter? A study investigated this question at a coffee bar in France. The waiter randomly assigned coffee-ordering customers into three groups: one received a card telling a joke with their bill, another group received a card containing an advertisement for a local restaurant, and the third received no card. The results are shown below.

-Use the Multiple Proportions applet to calculate the chi-square statistic and theory-based p-value for these data.
Chi-square statistic =
p-value =

Question 25

Can telling a joke affect whether or not a customer in a coffee bar leaves a tip for the waiter? A study investigated this question at a coffee bar in France. The waiter randomly assigned coffee-ordering customers into three groups: one received a card telling a joke with their bill, another group received a card containing an advertisement for a local restaurant, and the third received no card. The results are shown below.

-Use the Multiple Proportions applet to calculate the follow-up 95% confidence intervals for difference in proportions. Based on the output, which of the following statements are correct? Select all that apply.

A) The probability a customer leaves a tip is statistically significantly larger if a joke card is left than if an ad card is left.
B) The probability a customer leaves a tip is statistically significantly larger if a joke card is left than if no card is left.
C) The probability a customer leaves a tip is not statistically significantly different between the ad card and no card conditions.
D) The probability a customer leaves a tip is not statistically significantly different between the joke card and ad card conditions.

Question 26

As the chi-square statistic increases, the p-value increases.

Question 27

As the chi-square statistic increases, we have more evidence against the null hypothesis.

Question 28

The chi-square statistic can be negative.

Question 29

The chi-square statistic is the ratio of variability between the groups and the variability within the groups.

Question 30

What shape is the distribution of the chi-square statistic?

A) Symmetric
B) Normal
C) Right skewed
D) Left skewed

Question 31

Why do we do overall tests when comparing multiple proportions and not just do the follow-up confidence intervals?

A) Doing an overall test will more likely lead to significant results than just doing the follow-up confidence intervals.
B) Doing an overall test allows us to see exactly which group is significantly different from which other groups.
C) Doing an overall test allows us to quickly get our results, while the follow-up confidence intervals are very time-consuming.
D) Doing an overall test allows us to keep the probability of a type I error at 5% or whatever significance level we would like.

Question 32

Suppose we conduct a chi-square goodness-of-fit test to determine if a 6-sided die is fair. We roll the die 10 times, and the resulting p-value is very large. What may we conclude? Select all that apply.

A) We have proven that the die is fair.
B) The hypothesis that the die is fair is plausible.
C) There is evidence that the die is fair.
D) There is evidence against the hypothesis that the die is not fair.
E) There is little to no evidence that the die is not fair. F. We have proven that the hypothesis that the die is not fair is false.

Question 33

Are people equally likely to be born on any day of the seven days of the week? Or are some days more likely to be a person's birthday than other days? To investigate this question, days of birth were recorded for the 147 "noted writers of the present" listed in The World Almanac and Book of Facts 2000. The counts for the seven days of the week are given in the following table.

-Let ?_day be the probability that a person is born on a particular day. What is the null hypothesis?

A)
B)
C)
D)

Question 34

Are people equally likely to be born on any day of the seven days of the week? Or are some days more likely to be a person's birthday than other days? To investigate this question, days of birth were recorded for the 147 "noted writers of the present" listed in The World Almanac and Book of Facts 2000. The counts for the seven days of the week are given in the following table.
-Fill in the table below with the expected counts based on the hypothesized model that people are equally likely to be born on any day of the seven days of the week.

Question 35

Are people equally likely to be born on any day of the seven days of the week? Or are some days more likely to be a person's birthday than other days? To investigate this question, days of birth were recorded for the 147 "noted writers of the present" listed in The World Almanac and Book of Facts 2000. The counts for the seven days of the week are given in the following table.

-Use the Goodness of Fit applet to conduct a simulation-based chi-square goodness-of-fit test using the MAD (mean absolute difference) statistic, and report the statistic and the p-value. Use at least 1000 shuffles.
MAD statistic =
p-value =

Question 36

Are people equally likely to be born on any day of the seven days of the week? Or are some days more likely to be a person's birthday than other days? To investigate this question, days of birth were recorded for the 147 "noted writers of the present" listed in The World Almanac and Book of Facts 2000. The counts for the seven days of the week are given in the following table.

-Use the Goodness of Fit applet to conduct a simulation-based chi-square goodness-of-fit test using the chi-square statistic, and report the statistic and the p-value. Use at least 1000 shuffles.
chi-square statistic =
p-value =

Question 37

Are people equally likely to be born on any day of the seven days of the week? Or are some days more likely to be a person's birthday than other days? To investigate this question, days of birth were recorded for the 147 "noted writers of the present" listed in The World Almanac and Book of Facts 2000. The counts for the seven days of the week are given in the following table.

-Use the Goodness of Fit applet to conduct a theory-based chi-square goodness-of-fit test, and report the chi-square statistic and the p-value.
chi-square statistic =
p-value =

Question 38

Are people equally likely to be born on any day of the seven days of the week? Or are some days more likely to be a person's birthday than other days? To investigate this question, days of birth were recorded for the 147 "noted writers of the present" listed in The World Almanac and Book of Facts 2000. The counts for the seven days of the week are given in the following table.

-Are validity conditions met to conduct a theory-based chi-square goodness-of-fit test?

A) Yes, since all observed counts are at least 10.
B) Yes, since we have at least 10 successes and 10 failures.
C) No, since the p-values in questions 35 and 37 were different.
D) No, since this was not a random sample.

Question 39

Are people equally likely to be born on any day of the seven days of the week? Or are some days more likely to be a person's birthday than other days? To investigate this question, days of birth were recorded for the 147 "noted writers of the present" listed in The World Almanac and Book of Facts 2000. The counts for the seven days of the week are given in the following table.

-Based upon a p-value of 0.53, what is the appropriate conclusion for the test?

A) We have strong evidence that people are equally likely to be born on any day of the seven days of the week.
B) We do not have strong evidence that people are equally likely to be born on any day of the seven days of the week.
C) We have strong evidence that people are not equally likely to be born on any day of the seven days of the week.
D) We do not have strong evidence that people are not equally likely to be born on any day of the seven days of the week.

Question 40

According to the 2019 U.S. Census Bureau estimates, 63.4% of the U.S. self-identifies as Non-Hispanic white, 15.3% as Hispanic and Latino, 13.4% as Black or African American, 5.9% as Asian, and 2.0% as Other. A random sample of 1000 students graduating from California colleges and universities resulted in the following data on self-identified race: You would like to determine if the distribution of self-identified race among California college and university students differs from the distribution of self-identified race in the U.S. overall.

-What is the null hypothesis? Fill in the value for each parameter.

Question 41

According to the 2019 U.S. Census Bureau estimates, 63.4% of the U.S. self-identifies as Non-Hispanic white, 15.3% as Hispanic and Latino, 13.4% as Black or African American, 5.9% as Asian, and 2.0% as Other. A random sample of 1000 students graduating from California colleges and universities resulted in the following data on self-identified race: You would like to determine if the distribution of self-identified race among California college and university students differs from the distribution of self-identified race in the U.S. overall.

-In the null hypothesis, what does the symbol ?_Race represent?

A) The proportion of the U.S. that self-identify as a certain Race
B) The proportion of Californians that self-identify as a certain Race
C) The proportion of all students graduating from California colleges and universities that self-identify as a certain Race
D) The proportion of the sample of 1000 students that self-identified as a certain Race

Question 42

According to the 2019 U.S. Census Bureau estimates, 63.4% of the U.S. self-identifies as Non-Hispanic white, 15.3% as Hispanic and Latino, 13.4% as Black or African American, 5.9% as Asian, and 2.0% as Other. A random sample of 1000 students graduating from California colleges and universities resulted in the following data on self-identified race: You would like to determine if the distribution of self-identified race among California college and university students differs from the distribution of self-identified race in the U.S. overall.

-What is the expected count based on the null model for the "Hispanic or Latino" cell?

A) 77
B) 15.3
C) 11.8
D) 153

Question 43

According to the 2019 U.S. Census Bureau estimates, 63.4% of the U.S. self-identifies as Non-Hispanic white, 15.3% as Hispanic and Latino, 13.4% as Black or African American, 5.9% as Asian, and 2.0% as Other. A random sample of 1000 students graduating from California colleges and universities resulted in the following data on self-identified race: You would like to determine if the distribution of self-identified race among California college and university students differs from the distribution of self-identified race in the U.S. overall.

-Use the Goodness of Fit applet to conduct a simulation-based chi-square goodness-of-fit test using the chi-square statistic, and report the statistic and the p-value. Use at least 1000 shuffles.
chi-square statistic =
p-value =

Question 44

According to the 2019 U.S. Census Bureau estimates, 63.4% of the U.S. self-identifies as Non-Hispanic white, 15.3% as Hispanic and Latino, 13.4% as Black or African American, 5.9% as Asian, and 2.0% as Other. A random sample of 1000 students graduating from California colleges and universities resulted in the following data on self-identified race: You would like to determine if the distribution of self-identified race among California college and university students differs from the distribution of self-identified race in the U.S. overall.

-Use the
Goodness of Fit applet to conduct a simulation-based chi-square goodness-of-fit test using the MAD (mean absolute difference) statistic, and report the statistic and the p-value. Use at least 1000 shuffles.
MAD statistic =
p-value =

Question 45

According to the 2019 U.S. Census Bureau estimates, 63.4% of the U.S. self-identifies as Non-Hispanic white, 15.3% as Hispanic and Latino, 13.4% as Black or African American, 5.9% as Asian, and 2.0% as Other. A random sample of 1000 students graduating from California colleges and universities resulted in the following data on self-identified race: You would like to determine if the distribution of self-identified race among California college and university students differs from the distribution of self-identified race in the U.S. overall.

-Based upon a p-value of less than 0.001, what is an appropriate conclusion for the goodness-of-fit test?

A) We do not have strong evidence that the distribution of self-identified race among California college and university students differs from the distribution of self-identified race in the U.S. overall.
B) We have strong evidence that the distribution of self-identified race among California college and university students differs from the distribution of self-identified race in the U.S. overall.
C) We have strong evidence that the distribution of self-identified race among California college and university students is the same as the distribution of self-identified race in the U.S. overall.
D) We have strong evidence that the probabilities of each self-identified race among California college and university students are not all equal.

Question 46

According to the 2019 U.S. Census Bureau estimates, 63.4% of the U.S. self-identifies as Non-Hispanic white, 15.3% as Hispanic and Latino, 13.4% as Black or African American, 5.9% as Asian, and 2.0% as Other. A random sample of 1000 students graduating from California colleges and universities resulted in the following data on self-identified race: You would like to determine if the distribution of self-identified race among California college and university students differs from the distribution of self-identified race in the U.S. overall.

-Are the validity conditions met for a theory-based chi-square goodness-of-fit test?

A) Yes, since 1000 is larger than 20.
B) Yes, since all of the expected counts are at least 10.
C) No, since the observed count for Other is less than 10.
D) Yes, since the p-values from both the simulation-based and theory-based tests are similar.