Question 1

Let Group 1 be the group who was told, “Each piece you add costs 10 cents.” Let  Group 2 be the group who was told, “Each piece you add costs 10 cents but  removing pieces is free.”

1. Part A: Which of the following is a more appropriate description of the populations  under investigation?
   1. a) Population 1 is composed of people who did not get reminded to remove pieces, and Population 2 is composed of people who did get reminded to  remove pieces.
   2. b) Population 1 is composed of the experiment’s participants who did not get  reminded to remove pieces, and Population 2 is composed of the  experiment’s participants who did get reminded to remove pieces.
2. Part B: Based on the numbers provided in the article (particularly in the paragraph  copied at the beginning of the assignment), what do [latex]p_{1}[/latex] and [latex]p_{2}[/latex] represent?
   1. a) The proportion of people in Population 1 and Population 2, respectively, who thought to remove the destabilizing block.
   2. b) The proportion of people in Population 1 and Population 2, respectively, who didn’t think to remove the destabilizing block.
3. Part C: Which of the following expresses the null hypothesis, [latex]H_{0}[/latex]?
   1. a) [latex]p_{1} - p_{2} > 0[/latex]
   2. b) [latex]p_{1} - p_{2} < 0[/latex]
   3. c) [latex]p_{1} - p_{2} = 0[/latex]
   4. d) [latex]p_{1} - p_{2} \neq 0[/latex]
4. Part D: Suppose that the researchers wanted to test the claim that the proportion of  people who remove a piece is different depending on whether people are  reminded to remove a piece or not. Which of the following expresses the  alternative hypothesis, [latex]H_{A}[/latex]?
   1. a) [latex]p_{1} - p_{2} > 0[/latex]
   2. b) [latex]p_{1} - p_{2} < 0[/latex]
   3. c) [latex]p_{1} - p_{2} = 0[/latex]
   4. d) [latex]p_{1} - p_{2} \neq 0[/latex]

Question 2

Now that we’ve established the null and alternative hypotheses, we need to check  that the necessary conditions for conducting a two-sample test of proportions have  been met.

1. Part A: Fill in the following table based on the information in the article.

   Symbol	Meaning	Value
   [latex]x_{1}[/latex]	Number of people  in Group 1 who thought to remove  the destabilizing  block
   [latex]n_{1}[/latex]	Size of Group 1
   [latex]\hat{p_{1}}[/latex]	Proportion of people in Group 1  who thought to remove the destabilizing block
   [latex]x_{2}[/latex]	Number of people  in Group 2 who thought to remove  the destabilizing  block
   [latex]n_{2}[/latex]	Size of Group 2
   [latex]\hat{p_{2}}[/latex]	Proportion of people in Group 2  who thought to remove the destabilizing block
   [latex]\hat{p_{c}}[/latex]	Combined sample  proportion from both groups

   Recall the necessary conditions for a two-sample test of proportions.

   Conditions for Two-Sample Z-Test of Proportions

   1. Large Counts: Check that *MISSING LATEX*

   2. Random Samples/Assignment: Check that the two samples

   are independent and random samples or that they come from

   randomly assigned groups in an experiment.

   3. 10%: Check that *MISSING LATEX*.
2.  Part B: Verify Condition 2. Which of the following statements best addresses  whether or not this condition was satisfied?
   1. a) The article states that study participants were randomly assigned to either  Group 1 or Group 2, so this condition was satisfied.
   2. b) The article does not explicitly state that participants were randomly  assigned to either Group 1 or Group 2, but we know that the researchers  conducted an experiment, so we can safely assume that this condition  was satisfied.
   3. c) The article states that the first 98 people were assigned to Group 1 and  the next 99 people were assigned to Group 2, so this condition was not  satisfied.

   Part C: Verify Condition 3. Which of the following best expresses what is true about  this condition?

   1. a) We only need to check this condition if we are sampling from the population. Since this experiment was done with random assignment, we  do not need to check this condition.

    

   1. b) This condition was met.
3. Part D: Finally, we need to check that we have a large enough sample size to meet Condition 1. Using the table you filled in for Part A of this question, complete the following table.

   [latex]n_{1}\hat{p_{c}}[/latex]
   [latex]n_{1}(1-\hat{p_{c}})[/latex]
   [latex]n_{2}\hat{p_{c}}[/latex]
   [latex]n_{2}(1-\hat{p_{c}})[/latex]

   Part E: Is the sample size large enough to meet the “large counts” condition?

   1. a) Yes, we found that all values are greater than or equal to 10.
   2. b) No, there are some values that are less than 10.

Question 3

Now, we are ready to perform the test. We will use significance level α = 0.05. Go to  the DCMP Compare Two Population Proportions tool at

https://dcmathpathways.shinyapps.io/2sample_prop/.

Under “Enter Data,” select “Number of Successes.” For Group 1 and Group 2, enter  the appropriate values and check “Provide Group Labels” to add descriptions for each group.

1. Part A: Under “Type of Inference,” select “Significance Test.” Based on the  alternative hypothesis, which option should you select?
   1. a) Two-sided
   2. b) Less
   3. c) Greater
2. Part B: What is the observed difference of sample proportions?
3. Part C: Select the alternative hypothesis option you chose in Part A. What is the value of the z-test statistic?
4. Part D: What P-value do you obtain?

Question 4

In this question, you will interpret the results of the test using both the z-test statistic  and the P-value.

1. Part A: Which of the following is an appropriate interpretation of the z-test statistic?
   1. a) Our observed difference of sample proportions (−198) lies 2.78 standard  errors below the null hypothesis value. Since this lies more than 2  standard errors away, we know this value is quite unlikely, so we have  evidence to doubt the null hypothesis.
   2. b) Our observed difference of sample proportions (−198) lies 2.78 standard  errors above the null hypothesis value. Since this lies more than 2  standard errors away, we know this value is very likely, so we don’t have  evidence to doubt the null hypothesis.
   3. c) If we assume the null is true, there is a probability of 2.78 of seeing a  sample difference of proportions of −198 or more by chance alone. This  is very unlikely under the null, so we have reason to doubt the null  hypothesis.
   4. d) There is a probability of 2.78 that the null hypothesis is true.
2. Part B: Which of the following is an appropriate interpretation of the P-value?
   1. a) Our observed difference of sample proportions (–0.198) lies 0.0055 standard errors below the null hypothesis value. Since this lies less than 2  standard errors away, we know this value is quite unlikely, so we have  evidence to doubt the null hypothesis.
   2. b) Our observed difference of sample proportions (−198) lies 0.0055 standard errors above the null hypothesis value. Since this lies less than 2  standard errors away, we know this value is very likely, so we don’t have  evidence to doubt the null hypothesis.
   3. c) If we assume the null is true, there is a probability of 0.0055 of seeing a  sample difference of proportions of −198 or more by chance alone. This  is very unlikely under the null, so we have reason to doubt the null  hypothesis.
   4. d) There is a probability of 0.0055 that the null hypothesis is true.
3. Part C: Fill in the blank to express your conclusion.
   Under my assumption that there is no difference in proportions of people  who chose to remove Lego pieces, the observed data (a difference of  −0.198 between the two groups among 197 participants) is highly unlikely.
   Therefore, I _______ the assumption that there is no difference between the  two groups. There is evidence that the proportion of people who remove a  piece is different depending on whether people are reminded to remove a  piece or not.
   1. a) reject
   2. b) fail to reject

Question 5

In the next class, you will need to be able to use the data analysis tool to find a  confidence interval for the difference between two proportions. To do this, under  “Type of Inference,” select “Confidence Interval.” Construct a 95% confidence  interval.

1. Part A: What are the lower and upper bounds of the confidence interval you obtain  for [latex]p_{1} - p_{2}[/latex]? Fill in the following table and round to 3 decimal places.

   Lower bound
   Upper bound

2. Part B: Determine whether this statement is true or false: “There is a 95% chance  that the true difference in population proportions lies between −0.335 and  −0.061.”
   1. a) True
   2. b) False