Constructing Confidence Intervals for μ1-μ2. When the sampling distribution for x̅1-x̅2 is approximated by a t-distribution and the populations have equal variances, you can construct a confidence interval for μ1-μ2, as shown below.
Construct the indicated confidence interval for μ1-μ2 . Assume the populations are approximately normal with equal variances.
Family Doctor 
To compare the mean number of days spent waiting to see a family doctor for two large cities, you randomly select several people in each city who have had an appointment with a family doctor. The results are shown at the left. Construct a 90% confidence interval for the difference in mean number of days spent waiting to see a family doctor for the two cities. (Adapted from Merritt Hawkins)

Verified step by step guidance

1. Step 1: Identify the given data for both cities. For Miami, the sample mean \( \bar{x}_1 = 28 \) days, sample standard deviation \( s_1 = 39.7 \) days, and sample size \( n_1 = 20 \). For Seattle, the sample mean \( \bar{x}_2 = 26 \) days, sample standard deviation \( s_2 = 42.4 \) days, and sample size \( n_2 = 17 \).
2. Step 2: Since the populations are assumed to have equal variances, 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 standard error (SE) of the difference between the two sample means using the pooled standard deviation: \[ SE = s_p \times \sqrt{ \frac{1}{n_1} + \frac{1}{n_2} } \]
4. Step 4: Determine the degrees of freedom (df) for the t-distribution, which is \( n_1 + n_2 - 2 \). Then, find the critical t-value \( t^* \) corresponding to a 90% confidence level and the calculated degrees of freedom from the t-distribution table.
5. Step 5: Construct the 90% confidence interval for the difference in means \( \mu_1 - \mu_2 \) using the formula: \[ (\bar{x}_1 - \bar{x}_2) \pm t^* \times SE \] This interval estimates the range in which the true difference in mean waiting days between the two cities lies with 90% confidence.

Key Concepts

Confidence Interval for the Difference of Means

A confidence interval for the difference between two population means (μ1 - μ2) estimates the range within which the true difference lies with a certain level of confidence, such as 90%. It uses sample means, standard deviations, and sample sizes to calculate this range, providing insight into whether the means differ significantly.

t-Distribution with Equal Variances

When population variances are assumed equal but unknown, the sampling distribution of the difference between sample means follows a t-distribution. This approach pools the sample variances to estimate a common variance, which is then used to calculate the standard error and critical t-value for the confidence interval.

Assumptions of Normality and Equal Variances

Constructing a confidence interval using the t-distribution requires that the populations are approximately normal and have equal variances. These assumptions ensure the validity of the pooled variance estimate and the accuracy of the confidence interval, especially important when sample sizes are small or moderate.