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AP Statistics 8.5 Setting Up a Chi-Square Test for Homogeneity or Independence - MCQs - Exam Style Questions

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Question

A controversial issue in the sport of professional soccer is the use of instant replay for making difficult goal line decisions. Each person in a representative sample of \(102\) players, fans, coaches, and officials was asked his or her opinion about the use of instant replay for goal-line decisions. The data are summarized in the two-way frequency table below.
$$ \begin{array}{|l|c|c|} \hline \textbf{Category} & \textbf{Favor Use} & \textbf{Oppose Use} \\ \hline \textbf{Players} & 22 & 2 \\ \textbf{Fans} & 18 & 6 \\ \textbf{Coaches} & 15 & 26 \\ \textbf{Officials} & 10 & 3 \\ \hline \end{array} $$
In testing to see whether opinion with respect to the use of instant replay is independent of the category of the person interviewed, a chi-square test statistic of \(27.99\) and a p-value less than \(0.001\) were calculated. Which of the following statements is correct?
(A) The number of degrees of freedom for the test is \(8-1=7\).
(B) The chi-square test should not have been used because two of the counts in the table are less than \(5\).
(C) The null hypothesis states that there is an association between category and opinion about the use of instant replay, and the small p-value suggests that the null hypothesis should be rejected.
(D) The small p-value suggests that there is evidence of an association between category and opinion about the use of instant replay.
(E) The chi-square test shows that fans favor the use of instant replay.
▶️ Answer/Explanation
Detailed solution

1. State the Hypotheses for the Test:
– The null hypothesis (\(H_0\)) for a chi-square test of independence is that there is **no association** between the two variables.
– The alternative hypothesis (\(H_a\)) is that there **is an association**.

2. Interpret the P-value:
A small p-value (in this case, \(p < 0.001\)) indicates that the observed data would be very unlikely if the null hypothesis were true. Since the p-value is less than any standard significance level (like \(0.05\)), we reject the null hypothesis.

3. Make a Conclusion:
By rejecting the null hypothesis of no association, we conclude that there is convincing statistical evidence of an association between a person’s category (player, fan, etc.) and their opinion on the use of instant replay.
✅ Answer: (D)

Question

A survey was conducted in which both men and women were asked a question about a current issue. Possible responses to this question were “in favor of,” “not in favor of,” or “no opinion.” A chi-square test is to be used to determine whether the response to this question is independent of gender. The number of degrees of freedom for the chi-square test in this situation is
(A) 6
(B) 5
(C) 3
(D) 2
(E) 1
▶️ Answer/Explanation
Detailed solution

1. Identify Table Dimensions:
Gender: 2 categories (men, women)
Response: 3 categories (in favor, not in favor, no opinion)

2. Apply Chi-square Formula:
Degrees of freedom = \((r-1)(c-1)\) where \(r\) = rows, \(c\) = columns
Here: \((2-1)(3-1) = (1)(2) = 2\)

✅ Answer: (D) 2

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