Question

A market researcher for an automobile company suspects differences in preferred color between male and female buyers. Advertisements targeted to different groups should take such differences into account, if they exist. The researcher examines the most recent sales information of a particular car that comes in three colors. (You may find it useful to reference the appropriate table: chi-square table or F table)

	Sex of Automobile Buyer
Color	Male	Female
Silver	482	278
Black	546	310
Red	479	357

a. Choose the competing hypotheses to determine whether color preference depends on the automobile buyer’s sex.

• H₀: Color preference is independent of the automobile buyer’s sex.; H_A: Color preference is dependent on the automobile buyer’s sex.

• H₀: Color preference is dependent on the automobile buyer’s sex; H_A: Color preference is independent of the automobile buyer’s sex.

b-1. Calculate the value of the test statistic. (Round the intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

	Test statistic

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

H₀: Color preference and automobile buyer’s sex are independent.
H_a: Color preference and automobile buyer’s sex are not independent. .

b)

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square test for independence.

Analyze sample data. Applying the chi-square test for independence to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = (r - 1) * (c - 1) = (2 - 1) * (3 - 1)
D.F = 2
E_r,c = (n_r * n_c) / n

$\chi ^{2}=\sum \left [\frac{(O_{r,c}-E_{r,c})^{2}}{E_{r,c}} \right ]$
Χ² = 9.3052

where DF is the degrees of freedom.

The P-value is the probability that a chi-square statistic having 2 degrees of freedom is more extreme than 9.3052

We use the Chi-Square Distribution Calculator to find P(Χ² > 9.3052) = 0.0111.

Interpret results. Since the P-value (0.0111) is less than the significance level (0.05), we cannot accept the null hypothesis. Thus, we conclude that there is a relationship between Color preference and automobile buyer’s sex.