Question

A data set includes data from student evaluations of courses. The summary statistics are n=88​, x=4.07​, s=0.53. Use a 0.05 significance level to test the claim that the population of student course evaluations has a mean equal to 4.25. Assume that a simple random sample has been selected. Identify the null and alternative​ hypotheses, test​ statistic, P-value, and state the final conclusion that addresses the original claim. What are the null and alternative​ hypotheses?

Answer 1

Let be true mean value of student course evaluations. We want to test the claim that the population of student course evaluations has a mean equal to 4.25. This will be the null hypothesis and the alternative will be mean is not equal to 4.25

The hypotheses are

We know the following sample information

n=88 is the sample size

is the sample mean course evaluation

is the sample standard deviation of course evaluation

We will estimate the population standard deviation of course evaluation using the sample.

The standard error of mean is

is the hypothesized mean of student course evaluations (from the null hypothesis).

Since the sample size n is greater than 30, we can use normal distribution to approximate the distribution of sample mean. That means we will be using z statistics to test the hypotheses.

The test statistics is

This is a 2 tailed test (because the alter native hypothesis has "not equal to")

The p-value is the sum P(Z<-3.19)+P(Z>3.19)

Using the the standard normal tables we get for z=3.19, P(Z<3.19) = 0.5+0.4993=0.9993

Using P(Z>3.19) = 1-P(Z<3.19) = 1-0.9993=0.0007

Next we know that P(Z<-3.19) = P(Z>3.19)=0.0007, due to the symmetry of normal distribution.

Hence the p-value=0.0007+0.0007=0.0014

We will reject the null hypothesis if the p-value is less than the level of significance alpha=0.05.

Here, the p-value is 0.0014 and it is less than the significance level. Hence we reject the null hypothesis.

We conclude that there is sufficient evidence to support the claim that the population of student course evaluations has a mean not equal to 4.25.