University officials hope that changes they have made have improved the retention rate. Last year, a sample of 1937 freshmen showed that 1537 returned as sophomores. This year, 1575 of 1915 freshmen sampled returned as sophomores. Determine if there is sufficient evidence at the 0.05
level to say that the retention rate has improved. Let last year's freshmen be Population 1 and let this year's freshmen be Population 2.

a. State the null and alternative hypotheses for the test. Fill in the blank below.
b. Compute the value of the test statistic. Round your answer to two decimal places.
c. Make the decision and state the conclusion in terms of the original question.

Construct a confidence interval for the ratio of the two population variances using the given information. Assume the samples come from populations that are approximately normally distributed. Write your answer using interval notation, and round the interval endpoints to four decimal places.

n1=10 , n2=11, s21=7.324, s22=3.601, 95% level of confidence

A technician compares repair costs for two types of microwave ovens (type I and type II). He believes that the repair cost for type I ovens is greater than the repair cost for type II ovens. A sample of 69 type I ovens has a mean repair cost of $71.66. The population standard deviation for the repair of type I ovens is known to be $24.27. A sample of 69 type II ovens has a mean repair cost of $68.79. The population standard deviation for the repair of type II ovens is known to be $13.63. Conduct a hypothesis test of the technician's claim at the 0.1 level of significance. Let μ1 be the true mean repair cost for type I ovens and μ2 be the true mean repair cost for type II ovens.

a. State the null and alternative hypotheses for the test.
b. Compute the value of the test statistic. Round your answer to two decimal places.
c. Determine the decision rule for rejecting the null hypothesis H0. Round the numerical portion of your answer to two decimal places.
d. Make the decision for the hypothesis test.

Insurance Company A claims that its customers pay less, on average, than customers of its competitor, Company B. You wonder if this is true, so you decide to compare the average monthly costs of similar insurance policies from the two companies. For a random sample of 20 people who buy insurance from Company A, the mean cost is $178 per month with a standard deviation of $9. For 25 randomly selected customers of Company B, you find that they pay a mean of $185 per month with a standard deviation of $12
. Assume that the population distributions are approximately normal and the population variances are not equal.

Find the P
-value for the hypothesis test. Round your answer to four decimal places.

Adrian hopes that his new training methods have improved his batting average. Before starting his new regimen, he was batting 0.250 in a random sample of 56 at bats. For a random sample of 25 at bats since changing his training techniques, his batting average is 0.440. Determine if there is sufficient evidence to say that his batting average has improved at the 0.02
level of significance. Let the results before starting the new regimen be Population 1 and let the results after the training be Population 2.

a. State the null and alternative hypotheses for the test. Fill in the blank below.
b. Compute the value of the test statistic. Round your answer to two decimal places.
c. Make the decision and state the conclusion in terms of the original question.