Exercises. 1. For each of the following examples of tests of hypothesis about , show the rejection and nonrejection regions on the t distribution curve.

1. For each of the following examples of tests of hypothesis about , show the rejection and nonrejection regions on the t distribution curve.

a) A two tailed test with and

b) A left tailed test with and

c) A right tailed test with and

2. Consider the null hypothesis about the mean of a population that is normally distributed. Suppose a random sample of 20 observations is taken from this population to make this test. Using show the rejection and nonrejection regions and find critical value(s) for t for

a) left tailed test; b) two tailed test; c) right tailed test

3. Consider versus for a population that is normally distributed.

a) A random sample of 16 observations taken from this population produced a sample mean of 45 and a standard deviation of 5. Using , would you reject the null hypothesis?

b) Another random sample of 16 observations taken from the same population produced a sample mean of 41.9 and a standard deviation of 7. Using , would you reject the null hypothesis?

Comment on the result of parts a) and b).

4. Assuming that respective populations are normally distributed, make the following hypothesis tests.

a) ; ; ; ; ;

b) ; ; ; ; ;

c) ; ; ; ; ;

5. A business school claims that students who complete a three month course of typing course can type on average, at least 1200 words an hour.

A random sample of 25 students who completed this course typed, on average, 1130 words an hour with a standard deviation of 85 words. Assume that the typing speeds for all students who complete this course have an approximate normal distribution.

Using the 5% significance level, can you conclude that the claim of the business school is true?

6. The supplier of home heating furnaces of a new model claims that the average efficiency of the new model is at least 60. Before buying these heating furnaces, a distributor wants to verify the supplier’s claim is valid. To this end, the distributor chooses a random sample of 9 heating furnaces of a new model and measures their efficiency. The data are

63; 72; 64; 69; 59; 65; 66; 64; 65

Determine the rejection region of the test with . Apply the test and state your conclusion.

7. A past study claims that adults spend an average of 18 hours a week on leisure activities. A researcher wanted to test this claim. He took a sample of 10 adults and asked them about the time they spend per week on leisure activities. Their responses (in hours) were as follows

14; 25; 22; 38; 16; 26; 19; 23; 41; 33

Assume that the time spent on leisure activities by all adults is normally distributed. Using the 5% significance level, can you conclude that the claim of earlier study is true?

8. According to the department of Labor, private sector workers earned, on average $354.32 a week in 2001. A recently taken random sample of 400 private sector worker showed that they earn, on average, $362.50 a week with a standard deviation of $72. Find p -value for the test with an alternative hypothesis that the current wean weekly salary of private sector workers is different from $354.32.

9. A manufacturer of a light bulbs claims that the mean life of these bulbs is at least 2500 hours. A consumer agency wanted to check whether or not this claim is true. The agency took a random sample of 36 such bulbs and tested them. The mean life for the sample was found to be 2447 hours with a standard deviation of 180 hours.

a) Do you think that the sample information supports the company’s claim?

Use .

b) What is the Type I error in this case? Explain. What is the probability of making this error?

c) Will your conclusion of part a) change if the probability of making a

Type I error is zero?

10. Given the eight sample observations 31, 29, 26, 33, 40, 28, 30, and 25, test the null hypothesis that the mean equals 35 versus the alternative that it does not. Let .

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