Exam 8: Hypothesis Testing

A fleet of rental cars - all the same make, model, and year - has a mean fuel efficiency of $24.3$ miles per gallon (mpg). A random sample of 41 cars are selected and the air filter of each is replaced with a new one. Let $\mu$ be the population mean fuel efficiency score that would occur if every car's air filter were replaced. The air filter change is deemed effective if $\mu > 24.3 \mathrm { mpg }$ . A test is made of $H _ { 0 } : \mu = 24.3$ versus $H _ { 1 } : \mu > 24.3$ . Assume that the air filter changes are effective but the conclusion is reached that the chan might not be effective. Which type of error, of any, has occurred?

The probability of a type I error is represented by which of the following symbols?

Which type of null hypothesis is used in the figure below?

The power of a test measures the sensitivity of the test to detect a real difference in parameters if one actually exists.

The following display from a TI-84 Plus calculator presents the results of a hypothesis test for a population proportion $p$ . State the null and alternate hypotheses.

In a simple random sample of size 64 , there were 32 individuals in the category of interest. It is desired to test $H _ { 0 } : p = 0.36$ versus $H _ { 1 } : p < 0.36$ . Do you reject $H _ { 0 }$ $0.05$ level?

Find the critical value for the following values of the significance level $\alpha$ , sample size $n$ , and alternate hypothesis $H _ { 1 }$ . $\alpha = 0.05 , n = 7 , H _ { 1 } : \mu < \mu _ { 0 }$

Thirty-eight members of a bowling league sign up for a program that claims to improve b scores. The participants bowl a set of five games before the program begins and a set of fi games again at the end to measure their improvement. The mean number of points improved (over the set of five games) was $\bar { x } = 10$ . Assume th standard deviation is $\sigma = 45$ and let $\mu$ be the population mean number of points improved for the set of five games. To determine whether the program is effective, a test is made of the hypotheses $H _ { 0 } : \mu = 0$ versus $H _ { 1 } : \mu > 0$ . Compute the value of the test statistic. wling

A recent survey indicated that the average amount spent for breakfast by business managers was $7.58 with a standard deviation of $0.42. It was felt that breakfasts on the West Coast were higher than $7.58. A sample of 81 business managers on the West Coast had an average breakfast cost of $7.65. Find the P-value for the test.

State whether the null hypothesis should be rejected on the basis of the given $P$ -value. $P \text {-value } = 0.028 , \alpha = 0.10 \text {, two-tailed test }$

When a researcher fails to reject the null hypothesis, this does not mean that there is not enough evidence to support the .

In a survey of 702 cigarette smokers, 56 of them reported that they have tried hypnosis therapy to try to quit smoking. Can you conclude that less than one-tenth of smokers have tried hypnosis therapy? Use the $\alpha = 0.05$ level of significance.

A recent report by the American Medical Association stated the average annual salary of psychiatrists is $189,121 with a population standard deviation of $26,975. A group of Hospital administrators randomly sampled 64 psychiatrists and found an average annual Salary of $198,630. The group claims that the average annual salary is actually higher Than what the American Medical Association reported. What is the P-value for this One-sided test?

The following output from MINITAB presents the results of a hypothesis test. Test of $\mathrm { mu } = 53$ vs. not $= 53$ The assumed standard deviation $10.3$ Test of $m u=53$ vs. not $=53$ The assumed standard deviation $10.3$ Mean SE Mean 95\% CI 44 49.00 1.277084 (4.88394387,9.890112) -2.576019 0.009995 What are the null and alternate hypotheses?

The following output from MINITAB presents the results of a hypothesis test for a population mean $\mu$ . Test of $m u = 46$ vs. not $= 46$ Mean StDev SE Mean 95\% CI T P 57 48.49 16.622683 2.201728 (44.07943,52.90057) 1.130930 0.2629 What is the value of $s ?$

The mean annual tuition and fees for a sample of 12 private colleges was $36,800 with a standard deviation of $5000. A dotplot shows that it is reasonable to assume that the Population is approximately normal. You wish to test whether the mean tuition and fees For private colleges is different from $33,700. Compute the value of the test statistic and state the number of degrees of freedom.

Thirty-seven members of a bowling league sign up for a program that claims to improve bowling scores. The participants bowl a set of five games before the program begins and : five games again at the end to measure their improvement. The mean number of points improved (over the set of five games) was $\bar { x } = 19$ . Assume thi standard deviation is $\sigma = 58$ and let $\mu$ be the population mean number of points improved for the set of five games. To determine whether the program is effective, a test is made of the hypotheses $H _ { 0 } : \mu = 0$ versus $H _ { 1 } : \mu > 0$ . Compute the $P$ -value.

A machine that fills beverage cans is supposed to put 12 ounces of beverage in each can. The standard deviation of the amount in each can is $0.12$ ounce. The machine is overhaul, with new components, and ten cans are filled to determine whether the standard deviation changed. Assume the fill amounts to be a random sample from a normal population. 12.14 12.05 12.27 11.89 12.06 12.14 12.05 12.38 11.92 12.14 Perform a hypothesis test to determine whether the standard deviation differs from $0.12$ o Use the $\alpha = 0.05$ level of significance.