Testing the Difference Between Two Means, (c) find the standardized test statistic t,
Assume the samples are random and independent, and the populations are normally distributed.
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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.

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Identify the sample statistics given: For Dogs (group 1), the sample mean \( \bar{x}_1 = 255 \), sample standard deviation \( s_1 = 30 \), and sample size \( n_1 = 16 \). For Cats (group 2), the sample mean \( \bar{x}_2 = 231 \), sample standard deviation \( s_2 = 28 \), and sample size \( n_2 = 18 \).

Since the population variances are assumed equal, calculate the pooled standard deviation \( s_p \) using the formula: \[ s_p = \sqrt{ \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} } \]

Calculate the standard error (SE) of the difference between the two means using the pooled standard deviation: \[ SE = s_p \times \sqrt{ \frac{1}{n_1} + \frac{1}{n_2} } \]

Compute the test statistic \( t \) for the difference between two means with equal variances: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} \]

Determine the degrees of freedom (df) for the test: \[ df = n_1 + n_2 - 2 \] Use this \( t \)-value and \( df \) to compare against the critical value from the \( t \)-distribution at \( \alpha = 0.05 \) for a one-tailed test to decide whether to support the magazine's claim.

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Two-Sample t-Test for Means

This test compares the means of two independent groups to determine if there is a statistically significant difference between them. It assumes the samples are random, independent, and populations are normally distributed. The test statistic follows a t-distribution when population variances are unknown but assumed equal.

Pooled Variance and Equal Variance Assumption

When population variances are assumed equal, the sample variances are combined to calculate a pooled variance. This pooled variance provides a more accurate estimate of the common variance, which is used in the denominator of the t-test formula to standardize the difference between sample means.

Significance Level (α) and Hypothesis Testing

The significance level α (here 0.05) defines the threshold for rejecting the null hypothesis. If the calculated t-statistic falls in the critical region beyond this threshold, the null hypothesis is rejected, supporting the claim that one mean is greater than the other with a specified confidence level.