Exam 7: Inferences Based on a Single Sample: 355 Tests of Hypotheses

A revenue department is under orders to reduce the time small business owners spend filling out pension form ABC-5500. Previously the average time spent on the form was 5.7 hours. In order to test whether the time to fill out the form has been reduced, a sample of 86 small business owners who annually complete the form was randomly chosen, and their completion times recorded. The mean completion time for ABC-5500 form was 5.5 hours with a standard deviation of 2.3 hours. In order to test that the time to complete the form has been reduced, state the appropriate null and alternative hypotheses.

A random sample of $n$ observations, selected from a normal population, is used to test the null hypothesis $H _ { 0 } : \sigma ^ { 2 } = 155$ . Specify the appropriate rejection region. $H _ { \mathrm { a } } : \sigma ^ { 2 } \neq 155 , n = 10 , \alpha = .05$

A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 17% of women who actually have the disease. A new method has beendeveloped that researchers hope will be able to detect cancer more accurately. A random sample of 57 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in eleven. Specify the null and alternative hypotheses that the researchers wish to test.

The scores on a standardized test are reported by the testing agency to have a mean of 70. Based on his personal observations, a school guidance counselor believes the mean score is much higher. He collects the following scores from a sample of 50 randomly chosen students who took the test. 39 48 55 63 66 68 68 69 70 71 71 71 73 74 76 76 76 77 78 79 79 79 79 80 80 82 83 83 83 85 85 86 86 88 88 88 88 89 89 89 90 91 92 92 93 95 96 97 97 99 Use the data to conduct a test of hypotheses at $\alpha = .05$ .05 to determine whether there is any evidence to support the counselor's suspicions.