A study was conducted to determine if the salaries of elementary school teachers from two neighboring districts were equal. A sample of 15 teachers from each district was randomly selected. The mean from the first
District was $28,900 with a standard deviation of $2300. The mean from the second district was $30,300 with a
Standard deviation of $2100. Assume the samples are random, independent, and come from populations that
Are normally distributed. Construct a 95% confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ Assume that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \text {. }$
A)(-3047, 247)
B)(-2054, 238)
C)(-2871, 567)
D)(-4081, 597)

Question

Quizplus · Accepted Answer

The correct confidence interval is calculated using the formula for the difference between two means, assuming equal variances. Given the data, the formula for a 95% confidence interval for the difference between two means ($\mu_1 - \mu_2$) is $\bar{x}_1 - \bar{x}_2 \pm t_{\alpha/2} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$, where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $s_1^2$ and $s_2^2$ are the sample variances, $n_1$ and $n_2$ are the sample sizes, and $t_{\alpha/2}$ is the t-value for the desired level of confidence with degrees of freedom approximated by the smaller of $n_1-1$ and $n_2-1$. Given the means, standard deviations, and sample sizes, and assuming equal population variances, the calculation results in a 95% confidence interval that suggests the difference in means could range from -3047 to 247, indicating that there is not a significant difference between the two districts' teacher salaries at the 95% confidence level.

A Study Was Conducted to Determine If the Salaries of Elementary