Evaluating Model Fit Using Chi-Squared Tests

00:00 10 marks

A marketing company conducted a survey of 400 customers regarding their product preferences among four brands (A, B, C, D).

Part a

7 minutes 5 marks

Question

The company hypothesized that customer preferences would be uniformly distributed, meaning each product would be chosen by 25% of the customers. The observed frequencies for the products were as follows:

– Product A: 110
– Product B: 90
– Product C: 100
– Product D: 100

Perform a chi-squared goodness-of-fit test at the 5% significance level to determine whether the observed frequencies differ significantly from the expected uniform distribution.

For this test, state the null and alternative hypotheses, calculate the expected frequencies, compute the chi-squared statistic, determine the degrees of freedom, and compare the statistic with the appropriate critical value. Provide your conclusion regarding the hypotheses.

Hint Hint

Hint

For the goodness-of-fit test, remember that the expected frequency for each category under a uniform distribution is the total number of observations divided by the number of categories. Ensure you clearly state your hypotheses and show all calculation steps.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

Null hypothesis (\(H_0\)): Product preferences are uniformly distributed.
Alternative hypothesis (\(H_1\)): Product preferences are not uniformly distributed.
Expected frequency for each product = 400 * 0.25 = 100.
Chi-squared statistic (\(\chi^2\)) = 2.
Degrees of freedom (df) = 3.
Critical value at 5% significance level with df=3 is approximately 7.815.
Since 2 < 7.815, we fail to reject the null hypothesis. There is insufficient evidence to suggest that the observed product preferences differ significantly from a uniform distribution.

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet

Part b

8 minutes 5 marks

Question

In addition, the survey also collected data on customer gender. The observed frequencies for product preference by gender are given in the following contingency table:

Male:
– Product A: 60
– Product B: 40
– Product C: 30
– Product D: 20

Female:
– Product A: 50
– Product B: 50
– Product C: 70
– Product D: 80

Using a chi-squared test for independence at the 5% significance level, test whether there is an association between gender and product preference.

Hint Hint

Hint

For the chi-squared test for independence, remember to calculate expected frequencies for each cell in the contingency table using the formula: (row total * column total) / grand total. Clearly present your contingency table, expected frequencies, and all steps of the hypothesis test.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

Null hypothesis (\(H_0\)): Gender and product preference are independent.
Alternative hypothesis (\(H_1\)): Gender and product preference are associated.
Expected frequencies are calculated using (row total * column total) / grand total.
Chi-squared statistic (\(\chi^2\)) ≈ 31.33.
Degrees of freedom (df) = 3.
Critical value at 5% significance level with df=3 is approximately 7.815.
Since 31.33 > 7.815, we reject the null hypothesis. There is statistically significant evidence to suggest an association between gender and product preference.

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet