Testing the Difference Between Two Means, (e) interpret the decision in the context of the original claim. 
Assume the samples are random and independent, and the populations are normally distributed.
Transactions
 A magazine claims that the mean amount spent by a customer at Burger Stop is greater than the mean amount spent by a customer at Fry World. The results for samples of customer transactions for the two fast food restaurants are shown at the left. At , α=0.05 can you support the magazine’s claim? Assume the population variances are equal.

Verified step by step guidance

1. Step 1: Identify the hypotheses for the test. The magazine claims that the mean amount spent at Burger Stop (mean 1) is greater than the mean amount spent at Fry World (mean 2). So, the null hypothesis is \(H_0: \mu_1 \leq \mu_2\) and the alternative hypothesis is \(H_a: \mu_1 > \mu_2\).
2. Step 2: Since the population variances are assumed equal, calculate the pooled standard deviation using the formula: \[ s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \] where \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes.
3. Step 3: Calculate the test statistic (t) using the formula: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \] where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.
4. Step 4: Determine the degrees of freedom for the test, which is \(df = n_1 + n_2 - 2\), and find the critical t-value from the t-distribution table for a one-tailed test at \(\alpha = 0.05\).
5. Step 5: Compare the calculated t statistic to the critical t-value. If the test statistic is greater than the critical value, reject the null hypothesis and conclude that there is sufficient evidence to support the magazine's claim that the mean amount spent at Burger Stop is greater than at Fry World. Otherwise, do not reject the null hypothesis.

Key Concepts

Hypothesis Testing for Two Means

This involves comparing the means of two independent samples to determine if there is a statistically significant difference between them. The null hypothesis typically states that the means are equal, while the alternative hypothesis reflects the claim being tested, such as one mean being greater than the other.

Assumption of Equal Population Variances

When conducting a two-sample t-test, assuming equal population variances allows the use of a pooled variance estimate. This assumption simplifies calculations and affects the test statistic and degrees of freedom, making it important to verify or justify before proceeding.

Interpreting the Test Decision in Context

After performing the hypothesis test and comparing the p-value to the significance level (α=0.05), the decision to reject or fail to reject the null hypothesis must be interpreted in the context of the original claim. This means explaining what the statistical result implies about the magazine's claim regarding customer spending.