Teacher Salaries 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 random sample of 26 elementary school teachers is dollar 48,256, and the sample standard deviation is dollar 3,912.40 . The mean of the salaries of a random sample of 24 secondary school teachers is dollar 45,633 . The sample standard deviation is dollar 5533 . At $\alpha=0.05,$ 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.
in this problem. We're evaluating the claim that elementary school teachers make more money on average than secondary school teachers, so we have a no hypothesis when they start off with that. In fact, there is no difference that the average salary of elementary school teachers is the same as that of secondary school teachers. In this case are alternative hypothesis is that the average for elementary school teachers is greater and for secondary school teachers to evaluate this claim. Given the results of the sample of 26 elementary school teachers and 24 secondary school teachers, the X bar of sellers for the elementary school teachers is 48,526 $1000 per year, and the sample standard deviation of that sample is $3912. For the secondary school teachers. The X bar of the salaries was 45,006. 33 and there standard deviation of that sample was $5533. Obviously we were not given the sigma of the underlying population. This is a one sided test and our test statistic is going to be difference of the expires minus the difference in the expected values from over a standard error based on the sample standard deviation. This goes away because we're hypothesizing that the two means are equal. Therefore, hypothesize hypothesized differences zero that terms gonna go away and because we're using the sample standard deviations and because we are assuming that even though our sample sizes air less than 30 the underlying populations are normally distributed. Those two assumptions together allow us to assume that this test statistic is distributed according to the student T distribution, which requires that we calculate degrees of freedom. Member degrees of freedom is simply the sample size of the smaller of the two samples, which was 24 minus one. So we have 23 degrees of freedom. We are asked to evaluate the claim at ah significance level of Alfa equals 5% or 50.5 So if I have a now and were also asked to evaluate this using the probability value approach, So what? We're when we do that what we're basically saying is we asking the question, What is the probability of getting of the T statistic being greater than or equal to the number we actually got from the sample given degrees of freedom is 23 and given you 23 when we calculate the test statistic, we come up with one point nine to and with a T distribution with 23 degrees of freedom, the probability of getting a T statistic of 1.92 or greater is only 3.36% or 0.336 This is a smaller probability than the 0.5 or 5% statistic significance level we were given. And therefore we will reject the no hypothesis because if the no hypothesis were true, the probability of getting a test statistic of 1.92 would only be 3.36%. It would be highly improbable to get such a large test statistic if in fact the means were the same. Therefore, we're going to reject the null hypothesis that the mean salaries are the same. In a in favor of the alternative hypothesis that yes, elementary school teachers are paid more