a. Since we are given a normally distributed population, we need to use the z-test. The null and alternative hypotheses also tell us that the test would be two tailed (since it used equal and not equal to). To know the critical value, we look at a z-table.
ANSWER: The critical values at 10% level of significance is 1.645 and -1.645.
b-1. We can get the value of the test statistic by applying the following formula:
In the case of the problem, the population mean is equal to 410, the population standard deviation is 46, and the values of x and n are given in the problem. We just plug-in the values and calculate z.
b-2. To know if we are going to reject the null hypothesis or not, we just need to compare if the test statistic falls between -1.645 and 1.645. Since our test statistic is 2.20, this will be on the area which is greater than the critical value. Thus, we should be rejecting the null hypothesis.
ANSWER: Reject the null hypothesis. There is enough evidence to conclude that H is not equal to 410.
c. To determine the critical values at 5% level of significance, we refer to the z table again and look at the value under the two-tailed test and alpha of 0.05.
ANSWER: The critical values at 5% level of significance is 1.96 and -1.96.
d-1. To calculate the test statistic, we apply the same procedure as the one we did for subproblem b-1. We use the following formula:
For this subproblem, the population mean, population standard deviation, and n are still the same at 410, 46, and 85 respectively. The only thing that changed is the value of x. It's now at 397.
d-2. To know whether or not we should reject the null hypothesis, we need to do the same thing in subproblem b-2. The test statistic should be within -1.96 and 1.96 so that the null hypothesis will not be rejected. Unfortunately, -2.61 is smaller than -1.96 thus this will be in the area of rejection.
ANSWER: Reject the null hypothesis. There is enough evidence to conclude that H is not equal to 410.