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.




Mint dates of coins (years)


Philadelphia: x̅1=1984.8, s1=8.6


Denver: x̅2=1983.4, s2=8.4




Assume population variances are equal.

Verified step by step guidance

1. Identify the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \). Here, \( H_0: \mu_1 = \mu_2 \) (no difference in mean mint dates) and \( H_a: \mu_1 \neq \mu_2 \) (there is a difference in mean mint dates).
2. Since the population variances are assumed equal, use the pooled two-sample t-test formula. Calculate the pooled standard deviation \( s_p \) using the formula: \[ s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \] Note: You will need the sample sizes \( n_1 \) and \( n_2 \) for Philadelphia and Denver respectively.
3. Calculate the test statistic \( t \) using the formula: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \]
4. Determine the degrees of freedom \( df = n_1 + n_2 - 2 \) and use a t-distribution table or technology to find the critical t-value(s) for a two-tailed test at significance level \( \alpha = 0.05 \).
5. Compare the calculated \( t \)-statistic to the critical value(s). If \( |t| \) is greater than the critical value, reject \( H_0 \); otherwise, fail to reject \( H_0 \). Then, write a conclusion in context, stating whether there is sufficient evidence to conclude a difference in mean mint dates 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, variances, and 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 similar, allowing the use of a pooled variance estimate in the t-test. This assumption simplifies calculations and affects the degrees of freedom used. If variances differ greatly, a different 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, which states there is no difference between group means. If the p-value from the t-test is less than α, we reject the null, concluding a significant difference exists. Otherwise, we fail to reject it, indicating insufficient evidence.