Question

A researcher claims that the mean of the salaries of elementary school teachers is greater than the mean of the salaries of secondary school teachers in a large school district. The mean of the salaries of a sample of 26 elementary school teachers is $48,300, and the sample standard deviation is $3912.40. The mean of the salaries of a sample of 25 secondary school teachers is $46,200. The standard deviation is $5533. Ata = 0.10, can it be concluded that the mean of the salaries of the elementary school teachers is greater than the mean of the salaries of the secondary school teachers? Use the P-value method and use y, for the mean of the salaries for elementary school teachers. Assume the variables are normally distributed and the variances are unequal.

State the hypotheses and identify the claim with the correct hypothesis.
H0: μ1 ____ μ2 ____
H1: μ1 ____ μ2 ____
This hypothesis test is a _____ test.

Answer 1

This is a hypothesis test comparing the mean salaries of elementary and secondary school teachers using Welch's t-test. The null hypothesis is that the mean salary of elementary teachers is less than or equal to that of secondary teachers, and the alternative hypothesis is that the elementary teachers' mean salary is greater. It is a right-tailed test most appropriate for the scenario.

Step-by-step explanation:

The given scenario involves comparing the means of two independent samples from elementary and secondary school teachers to see if there is evidence that the mean salary of elementary school teachers is higher than that of secondary school teachers. Considering that we do not assume equal variances, we will conduct a two-sample t-test with unequal variances, also known as Welch's t-test.

Hypotheses

The null hypothesis (H0) postulates that there is no difference in the mean salaries: H0: μ1 ≤ μ2, where μ1 is the mean salary of elementary school teachers and μ2 is the mean salary of secondary school teachers.

The alternative hypothesis (H1) claims that the mean salary of elementary school teachers is greater than that of secondary school teachers: H1: μ1 > μ2. The claim made by the researcher corresponds with the alternative hypothesis.

Test Type

This hypothesis test is a right-tailed test because the claim is that one mean is greater than the other.

To conduct the test, we would use the sample means, sample standard deviations, and sample sizes to calculate the test statistic. Then, we'd find the corresponding p-value using a t-distribution with appropriate degrees of freedom as estimated by the Welch-Satterthwaite equation. A p-value less than the significance level (α = 0.10) indicates evidence against the null hypothesis in favor of the alternative.