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AP Statistics 3.14 Setting Up a Chi-Square Test for Homogeneity or Independence- Exam Style Questions - MCQs - New Syllabus

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Question

A curious customer wanted to determine if the distribution of types of miniature Hershey’s bars in variety packs are produced with equal distribution among the types. A random sample of Hershey’s miniature candy bars was taken, below is a table of the observed counts.

Assume all conditions for inference are met. Which of the following is the expected number of Krackle bars if the distribution among the types is equally distributed?

(A) 9.5
(B) 38
(C) 40
(D) 152
(E) 160

▶️ Answer/Explanation

To find the expected count under an equal distribution, first calculate the total number of candy bars:

\(44+46+38+32=160\)

Since there are 4 candy bar types and they are assumed to be equally distributed:

\(\text{Expected Count}=\frac{160}{4}=40\)

Therefore, the expected number of Krackle bars is 40 if all four candy bar types occur equally often.
✅ Answer: (C)

Question 

A cellphone company is tracking the color choices of its customers. One of the phones they sell has five different colors to choose from. In the past, four of the colors sold equally well, but one particular color was twice as popular as each of the others.

A random sample of 250 customers are surveyed, and their color preference is recorded.

Assuming any necessary conditions have been met, which significance test should be used to determine if the color choices continue to match the color ratios of the previous year?

(A) 1-proportion z-test
(B) 2-proportion z-test
(C) Chi-square test of homogeneity
(D) Chi-square test of independence
(E) Chi-square goodness-of-fit test

▶️ Answer/Explanation

The company wants to determine whether the current distribution of customer color preferences matches a previously known distribution. There is one categorical variable (phone color) with five categories and a specified set of expected proportions from the previous year.

A chi-square goodness-of-fit test is used when comparing observed counts from a single sample to expected counts based on a claimed population distribution.

A chi-square test of homogeneity compares distributions across multiple populations, and a chi-square test of independence examines the relationship between two categorical variables. Neither situation applies here.

✅ Answer: (E)

Question

Michelle believes there is a relationship between the number of months spent collecting baseball cards and whether the majority of the cards in a collector’s collection are regular or rare. Michelle took one random sample of \(500\) baseball card collectors from a national convention and organized the results in the two-way table shown below.
Majority Type of Baseball Cards and Months of Collecting Baseball Cards
 Fewer Than \(6\) Months\(6\)-\(10\) Months\(11\)-\(15\) Months\(16\)-\(20\) Months\(21\) or More MonthsTotal
Has a Majority of Regular Baseball Cards\(80\)\(84\)\(71\)\(76\)\(112\)\(423\)
Has a Majority of Rare Baseball Cards\(11\)\(16\)\(9\)\(6\)\(35\)\(77\)
Total\(91\)\(100\)\(80\)\(82\)\(147\)\(500\)
Which hypothesis test and conclusion are appropriate if Michelle obtains a \(p\)-value of \(0.0075\), assuming conditions for inference are met and using \(\alpha=0.05\)?
(A) Chi-square test for independence; reject \(H_0\) and conclude there is convincing evidence of an association between months collecting and majority card type.
(B) Chi-square test for homogeneity; fail to reject \(H_0\) and conclude there is no evidence of an association between months collecting and majority card type.
(C) Two-sample \(z\)-test for proportions; reject \(H_0\) and conclude that the proportion of regular cards is greater than the proportion of rare cards.
(D) One-sample \(z\)-test for a proportion; fail to reject \(H_0\) because \(0.0075<0.05\).
▶️ Answer/Explanation

Michelle took one random sample and measured two categorical variables: number of months collecting baseball cards and majority card type.
Therefore, the appropriate test is a chi-square test for independence.
The hypotheses are:
\(H_0\): There is no association between number of months collecting and majority card type for all baseball card collectors at the convention.
\(H_a\): There is an association between number of months collecting and majority card type for all baseball card collectors at the convention.
Since the \(p\)-value is \(0.0075\) and \(\alpha=0.05\),
\(0.0075<0.05\), so we reject \(H_0\).

There is convincing statistical evidence that there is an association between the number of months spent collecting baseball cards and whether the majority of cards in the collection are regular or rare.

✅ Answer: (A)

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