**1 Population**

**Hypothesis Testing**

Probability & Statistics (MA220)

In a frequently traveled stretch of the I-75 highway, where the posted speed is 70 mph, it is thought that people travel on the average of at least 75 mph. To check this claim, the following radar measurements of the speeds (in mph) is obtained for 10 vehicles traveling on this stretch of the interstate highway.

66, 74, 79, 80, 69, 77, 78, 65, 79, 81

Do the data provide sufficient evidence to indicate that the claim is true? Test the appropriate hypothesis using 𝛼 = 0.01.

Example 7.4

Population Variance Known

Small Sample

Population Variance Unknown

Large Sample

Binomial to Normal Approx.

A cheque-cashing service company found that approximately 7% of all cheques submitted to the service were without sufficient funds. After instituting a random check verification system to reduce its losses, the service company found that only 70 were rejected in a random sample of 1125 that were cashed. Is there sufficient evidence that the cheque verification system reduced the proportion of bad checks at α = 0.01? What is the p-value associated with the test? What would you conclude at the α = 0.05 level?

Example 7.6

Population Mean

Population Proportion

**2 Populations**

In a salary equity study of faculty at a certain university, sample salaries of 50 male assistant professors and 50 female assistant professors yielded the following basic statistics.

Test the hypothesis that the mean salary of male assistant professors is more than the mean salary of female assistant professors at this university.

Example 7.7

Population Proportions

Population Variances Known

Population Mean

Population Variances Unknown

Population Means

A new diet and exercise program has been advertised as remarkable way to reduce blood glucose levels in diabetic patients. Ten randomly selected diabetic patients are put on the program, and the results after 1 month are given by the following table.

Do the data provide sufficient evidence to support the claim that the new program reduces blood glucose level in diabetic patients? Use α = 0.05.

Example 7.8

2 Independent Populations

2 Dependent Populations

It is claimed that sports-car owners drive on the average 18,000 miles per year. A consumer firm believes that the average mileage is probably lower. To check, the consumer firm obtained information from 40 randomly selected sports-car owners that resulted in a sample mean of 17,463 miles with a sample standard deviation of 1348 miles. What can we conclude about this claim?

Example 7.5

The drug company and their cough syrup manufacturer are having a dispute.

The factory says that the amount of syrup that gets poured into their bottles follows a distribution X~N(355, 25), where X is the amount of syrup in the bottle measured in mL. The drug company conducted tests on a large sample and found that the mean amount of syrup in 100 bottles is 356.5 mL. Test the hypothesis that the factory mean is correct at a 5% level of significance against the alternative that the mean amount of syrup in a bottle is greater than 355 mL. Assume the claim about normal distribution is correct.

Example 7.3

Equal Variances

Unequal Variances