AP Statistics 3.14: Setting Up Chi-Square Tests

Set up a chi-square test by choosing homogeneity or independence and stating the appropriate null and alternative hypotheses.

Covered syllabus points

3.14.A—Describe chi-square distributions 3.14.B—Identify an appropriate testing method for comparing distributions in two-way tables of categorical data…3.14.C—Identify the null and alternative hypotheses for a chi-square test for homogeneity or independence 3.14.D—Justify the appropriateness of a chi-square test for independence or homogeneity by verifying conditions

[Maximum number: 1]

A marketing research consultant for a hotel chain hypothesizes that men and women differ in their color preference for guest rooms. The consultant shows pictures of rooms decorated with three different color schemes to each person in a random sample of 110 men and to each person in a random sample of 90 women. The consultant asks each person to choose his or her favorite color scheme. A chi-square test for homogeneity of proportions will be used to test the consultant's hypothesis. Assuming that the conditions for inference are met, which of the following statements is true for the test?

The null hypothesis for the test is that the proportion of each gender who prefer each color scheme is $\frac{1}{3}$ .

The sample size is too small to detect a significant difference in a chi-square test for homogeneity of proportions.

The test is not valid because the sample sizes are not equal.

The more that men and women differ in their color preferences, the larger the chi-square statistic will be.

The test would also be appropriate if 90 married couples had been used for the two samples.

[Maximum number: 4]

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.

	Fewer Than 6 Months	6-10 Months	11-15 Months	16-20 Months	21 or More Months	Total
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

Majority Type of Baseball Cards and Months of Collecting Baseball Cards

(a)

Michelle believes there is a relationship between the number of months spent collecting baseball cards and which type of card is the majority in the collection (regular or rare).

(i)

Name the hypothesis test Michelle should use to investigate her belief. Do not perform the hypothesis test.

(ii)

State the appropriate null and alternative hypotheses for the hypothesis test you identified in (c-i). Do not perform the hypothesis test.

Model Solution
$H_0$ : Months collecting cards and majority card status are independent for all baseball card collectors at the convention.\n $H_a$ : Months collecting cards and majority card status are not independent for all baseball card collectors at the convention.

Scoring components
2. States the correct null and alternative hypotheses.
3. Includes both months collecting and majority card status in context.

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Covered syllabus points

3.14 Setting Up a Chi-Square Test for Homogeneity or Independence question 1

3.14 Setting Up a Chi-Square Test for Homogeneity or Independence question 2

Question (a)

Question (i)

Question (ii)