AP Statistics 3.15 Carrying Out a Chi Square Test for Homogeneity or Independence Questions

Carry out a chi-square test for a two-way table by finding expected counts, calculating the statistic and p-value, and stating a contextual conclusion.

Exam points

calculate an expected cell count as row total times column total divided by the grand total
keep expected counts as model-based values and use them to calculate individual chi-square contributions
sum observed-minus-expected squared over expected across every cell in the two-way table
use rows minus one times columns minus one for homogeneity or independence degrees of freedom
interpret a right-tail p-value as conditional on the contextual homogeneity or independence null model
compare the chi-square p-value with alpha to decide whether to reject or fail to reject the null
state contextual evidence for association or differing distributions in the relevant population or populations
after significance, compare observed and expected cell patterns to describe what drives the association

Covered syllabus points

3.15.A—Calculate expected counts for two-way tables of categorical data 3.15.B—Calculate the appropriate test statistic and p-value for a chisquare test for homogeneity or independence 3.15.C—Interpret the p-value for the chi-square test for homogeneity or independence 3.15.D—Justify a claim about the population(s) based on the results of a chi-square test for homogeneity or independence

[Maximum number: 1]

Questions 6-8 refer to the following situation: A computer manufacturer sets up three locations to provide technical support for its customers. Logs are kept, noting whether or not calls about problems are solved successfully. Data from a sample of 1,000 calls are summarized in the following table:

Assuming there is no association between location and whether or not a problem is resolved successfully, what is the expected number of successful calls (problem solved) from location 1?

$\frac{(325)(450)}{700}$

$\frac{(450)(700)}{1,000}$

$\frac{(325)(700)}{450}$

$\frac{(325)(700)}{1,000}$

．Three months from now，all of the parents／guardians of elementary school students in a certain large school

district will vote either yes or no on a new electronic tablet program．This program will allow every elementary

school student within the school district to be issued a tablet．In a recent survey of a random sample of 300

parents／guardians of elementary school students， 146 out of 300 （48．7 percent）indicated they would vote yes in

favor of such a program．

Mr．Robinson，the technology expert at the school district，is in favor of the elementary students being issued a

tablet．He develops a pamphlet showing the educational benefits of students having their own tablet．He did this to

increase the proportion of parents／guardians voting yes．To investigate the effectiveness of the pamphlet，Mr．

Robinson distributed the pamphlet to the same 300 parents／guardians from the original sample．After reading the

pamphlet，each parent／guardian was asked again whether each elementary student should be issued a tablet．

Parents／guardians could change their answer or keep the same answer from the original survey．The results of the

parent／guardian replies are shown in the table．

(a)

（c）（i）Assume the conditions for McNemar＇s test have been met．Calculate the test statistic using the sample data

from Mr．Robinson＇s study．Show your work．

Use a pencil or a pen with black or dark blue ink．Do NOT write your name．Do NOT write outside the box．

Question 6

Continue your response to QUESTION 6 on this page．

\begin{aligned} & \text { Using } E_{i}=\frac{R_{i} g}{n}, \text { expected value has been calcenlated: } \\ & X^{2}=\frac{(110-89.113)^{2}}{110+89.113}+\frac{(36-58.886)^{2}}{36+58.886}+\frac{(69-91.886)^{2}}{69+91.886}+\frac{(85-62.113)^{2}}{85+62.113}=14.527 \end{aligned}

Use a pencil or a pen with black or dark blue ink．Do NOT write your name．Do NOT write outside the box．

Question 6

Continue your response to QUESTION 6 on this page．

(b)

Assuming the null hypothesis is true，McNemar＇s test statistic， $\chi^{2}$ ，is distributed approximately as a chi－square

statistic with 1 degree of freedom．Determine the p－value for Mr．Robinson＇s test using the test statistic found

in part（c－i）．

p－value for Mr．Robinson＇s test using the test statistic found

in part（c－i）．

p\left(x^{2}>14.527\right)=1.3817 \times 10^{-4}

（d）（i）Using your results in part（c），what should Mr．Robinson conclude for his hypothesis test at a significance

level of 0．05？

(c)

（d）（i）Using your results in part（c），what should Mr．Robinson conclude for his hypothesis test at a significance

level of 0．05？

(d)

Recall that Mr．Robinson＇s goal was to increase support for the tablet program．Using only the 105 responses

of the parents／guardians whose opinions changed，explain how Mr．Robinson might conclude the pamphlet

[Maximum number: 1]

A chi-square test of homogeneity was conducted to determine whether different racial groups have different distributions in the size of their social networks. The chi-square test statistic and the p-value of the test were 17.43 and 0.037 , respectively. Which of the following is the correct interpretation of the p-value?

Assuming that each racial group has the same distribution in size of social network, the probability of finding a test statistic of 17.43 or smaller is 0.037.

Assuming that each racial group has the same distribution in size of social network, the probability of finding a test statistic of 17.43 or larger is 0.037.

There is a 0.037 probability that the different racial groups have the same distribution of social networks.

There is a 0.037 probability that the different racial groups have different distributions of social networks.

Baseball cards are trading cards that feature data on a player's performance in baseball games. Michelle is at a national baseball card collector's convention with approximately 20,000 attendees. She notices that some collectors have both regular cards, which are easily obtained, and rare cards, which are harder to obtain. Michelle believes that there is a relationship between the number of months a collector has been collecting baseball cards and whether the majority of the cards (cards appearing more often) in their collection are regular or rare. She obtains information from a random sample of 500 baseball card collectors at the convention and records how many full months they have been collecting baseball cards and whether the majority of the cards in their card collection are regular or rare. Her results are displayed in a two-way table.

Majority Type of Baseball Cards and Months of Collecting Baseball Cards

After completing the hypothesis test described in part (c), Michelle obtains a p-value of 0.0075. Assuming the conditions for inference are met, what conclusion should Michelle make about her belief? Justify your response.

Because the p-value of 0.0075 is less than any
reasonable $\alpha$ level such as 0.05 or 0.10, the null
hypothesis should be rejected. The data provide
convincing statistical evidence that there is an association between months collecting baseball
cards and majority card status for all baseball
card collectors at the convention.

Essentially correct (E) if the response satisfies the following two components:
1. Provides correct comparison of the p-value to
alpha ( p-value is less than alpha) AND provides a
correct decision about the null and/or alternative
hypothesis
2. States a conclusion, in context, consistent with and in terms of the stated alternative hypothesis
in part (c) using nondefinitive language
Partially correct (P) if the response satisfies only one of the two components required for E.
Incorrect (I) if the response does not meet the
criteria for E or P.

Additional Notes

- Because no alpha value is explicitly given, a response may satisfy component 1 by stating that the p-value is small AND providing a correct decision about the null and/or alternative hypothesis.

- An explicit decision about the null hypothesis is not required to satisfy component 1.

- If an explicit decision is stated and the conclusion is inconsistent with the decision, component 1 is not satisfied.

- The decision part of component 1 may be satisfied by implying the decision within the conclusion statement (sufficient evidence/insufficient evidence for the alternative hypothesis).

- To satisfy the context in component 2 the response must include discussion of both "months spent collecting baseball cards" and "which type of card is the majority in the collection."

- If the response omits hypotheses in part (c), assume the correct alternative hypothesis is provided when scoring component 2.

- If the response states incorrect hypotheses in part (c), components 1 and 2 may be satisfied by either referring to the stated alternative hypothesis or referring to the inference question.

- Examples of nondefinitive language in component 2 include "evidence to accept the alternative," "there is evidence for the alternative," "there is not sufficient evidence for the alternative," and "Michelle should conclude that her belief is correct."

- Examples of definitive language in component 2 include "proves the null," "proves the alternative," "accepts the alternative," "there is not evidence for the alternative," "no evidence for the alternative," "therefore the alternative is true," and "there is an association."

- If components 1 and/or 2 are satisfied and the response provides an incorrect interpretation of the p-value, the score is lowered from E to P or P to I.

All question bank results loaded

Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question 3

Question 4