Question

Independent random samples taken at two local malls provided the following information regarding purchases by patrons of the two malls.

Hamilton Place Eastgate

Sample Size 85 93

Average Purchase $143 $150

Standard Deviation (s) $22 $18

We want to determine whether there is a significant difference between the average purchases by the patrons of the two malls.

a. Give the hypotheses for the above.

b. Compute the test statistic.

c. At a .05 level of significance, test the hypotheses.

d. What is the probability of not causing a Type I error?

e. What is the probability that you would case a Type II if there is a difference?

Please explain step by step and elaborate your answers in Question d and e.

Answer 1

The student's question involves performing a hypothesis test for two independent sample means to determine if there is a significant difference between the average purchases at two malls. A null and alternative hypothesis are stated, a test statistic is computed, and a decision is made based on the comparison of the p-value and alpha level, with probabilities for Type I and Type II errors discussed.

Step-by-step explanation:

Step-by-Step Explanation of Hypothesis Testing Regarding Average Purchases

To answer the student's question regarding whether there is a significant difference between the average purchases by the patrons of two malls, we need to perform a hypothesis test for two independent sample means. Here are the steps:

State the hypotheses: The null hypothesis (H0) is μ1 = μ2, which means there is no difference between the average purchases at the two malls. The alternative hypothesis (H1) is μ1 ≠ μ2, indicating there is a difference.

Compute the test statistic: The test statistic for two independent samples can be calculated using the formula for the t-test.

Make a Decision: Compare the computed t-test statistic to the critical value from the t-distribution at a 0.05 alpha level. If the test statistic is beyond the critical value, we reject H0, otherwise, we do not reject H0.

Determine the probability of a Type I error: The probability of a Type I error (α) is the level of significance, which is 0.05 or 5% in this case.

Calculating the probability of a Type II error is more complex and requires additional information such as the actual difference in means (effect size) and the power of the test. Without this information, we cannot provide a precise probability.

The decision is based on the comparison of the p-value and alpha. If p-value > alpha, we do not reject the null hypothesis, indicating there is insufficient evidence to say there is a significant difference between the average purchases. The probability of not making a Type I error is 1 - α (in this instance, 95%). As for the probability of a Type II error, it's not possible to calculate without specific power or effect size information.