In Exercises 4 and 5, use technology to perform a two-sample t-test to determine whether there is a difference in the mint dates and in the values of coins found on a street from 1985 through 1996 for the two mint locations. Write your conclusion as a sentence. Use α = 0.05.




Value of coins (dollars)


Philadelphia: x̅1=\(0.034, s1=\)0.054


Denver: x̅2=\(0.033, s2=\)0.052




Assume population variances are equal.

Verified step by step guidance

1. Identify the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). Here, \(H_0\): There is no difference in the mean values of coins from Philadelphia and Denver, i.e., \(\mu_1 = \mu_2\). \(H_a\): There is a difference, i.e., \(\mu_1 \neq \mu_2\).
2. Since the population variances are assumed equal, use the pooled two-sample t-test. Calculate the pooled standard deviation \(s_p\) using the formula: \[s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\] where \(n_1\) and \(n_2\) are the sample sizes for Philadelphia and Denver respectively.
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, and \(s_p\) is the pooled standard deviation.
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 two-tailed test at significance level \(\alpha = 0.05\).
5. Compare the calculated test statistic \(t\) with the critical t-value. If \(|t|\) is greater than the critical value, reject the null hypothesis; otherwise, fail to reject it. Then, write a conclusion in context, stating whether there is sufficient evidence to say the mean coin values differ between the two mint locations.

Key Concepts

Two-Sample t-Test

A two-sample t-test compares the means of two independent groups to determine if there is a statistically significant difference between them. It uses sample means, standard deviations, and sample sizes to calculate a t-statistic, which is then compared to a critical value based on the chosen significance level (α). This test assumes the data are approximately normally distributed.

Equal Population Variances Assumption

Assuming equal population variances means the variability in both groups is considered the same, allowing the use of a pooled variance estimate in the t-test. This assumption simplifies calculations and affects the degrees of freedom used. If this assumption is violated, a different version of the t-test (Welch’s t-test) is more appropriate.

Significance Level (α) and Hypothesis Testing

The significance level α (here 0.05) defines the threshold for rejecting the null hypothesis, representing a 5% risk of a Type I error. Hypothesis testing involves stating a null hypothesis (no difference) and an alternative hypothesis (difference exists), then using the test statistic and α to decide whether to reject the null and conclude a significant difference.