Testing the Difference Between Two Means, 
(d) decide whether to reject or fail to reject the null hypothesis. 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: State the hypotheses. The null hypothesis \(H_0\) is that the mean amount spent by customers at Burger Stop is equal to or less than that at Fry World, i.e., \(\mu_1 \leq \mu_2\). The alternative hypothesis \(H_a\) is that the mean amount spent at Burger Stop is greater, i.e., \(\mu_1 > \mu_2\).
2. Step 2: Since the population variances are assumed equal, calculate the pooled standard deviation \(s_p\) using the formula: \(s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\)
3. Step 3: Calculate the test statistic \(t\) using the formula: \(t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}\)
4. Step 4: Determine the degrees of freedom, which is \(df = n_1 + n_2 - 2\), and find the critical value \(t_{\alpha}\) for a one-tailed test at \(\alpha = 0.05\).
5. Step 5: Compare the calculated test statistic \(t\) to the critical value \(t_{\alpha}\). If \(t\) is greater than \(t_{\alpha}\), reject the null hypothesis; otherwise, fail to reject it. This will help decide if there is enough evidence to support the magazine's claim.

Key Concepts

Hypothesis Testing for Two Means

This involves comparing the means of two independent samples to determine if there is statistical evidence that the population means differ. The null hypothesis typically states that the means are equal, while the alternative reflects the claim, such as one mean being greater than the other.

Pooled Variance and Equal Population Variances Assumption

When population variances are assumed equal, a pooled variance estimate combines the sample variances to improve the accuracy of the test statistic. This assumption simplifies the calculation of the standard error and affects the degrees of freedom used in the t-test.

Significance Level and Decision Rule

The significance level (α) defines the threshold for rejecting the null hypothesis, commonly set at 0.05. If the calculated p-value is less than α, the null hypothesis is rejected, supporting the alternative claim. Otherwise, there is insufficient evidence to reject the null.