Assignment 5A. Theory Component

Show your work for this component. Where relevant and unless otherwise instructed, keep your calculations and your final answer to at least four decimals. For any test of hypothesis questions, show all steps in detail and provide a conclusion in terms of the context of the question.

Problem 1. Salaries (23 marks)

A study was conducted to determine if the salaries of the professors from two neighbouring universities were equal. A sample of 20 professors from each university was randomly selected. The mean from the first university was $109,100 with a population standard deviation of $2300. The mean from the second university was $110,500 with a population standard deviation of $2100. Assume that the distribution of professor salaries, at both universities, are approximately normally distributed. Level of significance = 0.05.

1. Using the critical value approach test the claim that the salaries from both universities are equal. Include all key steps of the test. [10 marks]
2. Use the four-step P-value approach and test the claim that the salaries from both universities are equal. [10 marks]
3. Are the conditions met for using this test? [3 marks]

Problem 2. Flexibility (12 marks)

Are women more flexible than men? To answer this question, a physical therapist measured the flexibility of 15 randomly selected women and 20 randomly selected men by determining the number of inches subjects could reach while sitting on the floor with their legs straight out and back perpendicular to the ground. The more flexible an individual is, the more they can reach from the seated position. The physical therapist obtained the following data:

Gender	Sample Size	Mean ( in inches)	SD (in inches)
Women	15	20.86	2.11
Men	20	18.43	3.16

Assume that both population distributions are approximately normal and that the population variances are NOT equal.

1. Is the sampling dependent or independent in this study? Justify your answer. [2 marks]
2. Test the hypothesis at = 0.05. Using the critical value approach, include all key steps of test of hypothesis. [10 marks]

Problem 3. Customer waiting times (10 marks)

Bank A claims that the waiting time for its customers is less than its competitor Bank B's customers. A researcher collected data on the wait time for customers from both banks. Test the claim. Assume the samples are random and independent, and the populations are normally distributed. Also, assume that the unknown population variances are equal. Test the claim at 0.01 level of significance.

Bank A	Bank B
= 15 = 5.3 minutes = 1.1 minutes	= 16 = 5.6 minutes = 1.0 minutes

Is there sufficient evidence to believe that the wait time for Bank A is less than for Bank B? Include all key steps of test of hypothesis, using the critical value approach. [10 marks]

Problem 4. Cholesterol levels (12 marks)

A pharmaceutical company introduced a new drug that would reduce cholesterol levels. A researcher selected 10 cholesterol patients and tested them for their cholesterol level. The subjects were then placed on the new drug for a period of 6 months, and the cholesterol level reading was taken again. Both the results are given below. Assume the samples are random and dependent, and the populations are normally distributed.Use 0.01 level of significance and test whether there is evidence of reduced cholesterol level.

Subject	A	B	C	D	E	F	G	H	I	J
Before	196	224	201	196	176	259	265	193	240	268
After	181	219	209	176	170	229	255	190	240	250

1. Is the sampling dependent or independent in this study? Justify your answer. [2 marks]
2. Use the critical value approach, test whether there is evidence of reduced cholesterol level. Include all key steps of the test. [10 marks]

Problem 5. Health care (23 marks)

In a random survey of 500 doctors that practice specialized medicine, 20% felt that health care should be privatized in Canada. In a random sample of 800 family physicians, 30% felt that the health care should be privatized in Canada. Is there any difference in the two proportions? Use = 0.10.

1. Are the conditions met for using a z test to test this claim? [3 marks]
2. Use the four-step P-value approach and test the claim described here. [10 marks]
3. Using the critical value approach, test the hypothesis for the claim described here. Include all key steps of the test of hypothesis. [10 marks]