Exam 8: Hypothesis Testing

Scores on an IQ test are normally distributed. A sample of 17 IQ scores had standard deviation $s = 23$ . The developer of the test claims that the population standard deviation is $\sigma = 17$ . Do these data provide sufficient evidence to contradict this claim? Use the $\alpha = 0.01$ level of significance.

Sam Ying, a career counselor, claims the average number of years of schooling for an engineer is 15.8 years. A sample of 16 engineers had a mean of 15.0 years and a standard deviation of 1.5 years. The test value used in evaluating the claim would be -2.68.

A doctor claims that the standard deviation of systolic blood pressure is $12 \mathrm { mmHg }$ . A random sample of 24 patients found a standard deviation of $14 \mathrm { mmHg }$ . Assume the variable is normally distributed. At $\alpha = 0.01$ , what are the critical $\chi ^ { 2 }$ values?

Reginald Brown, an inspector from the Department of Weights and Measures, weighed 15 eighteen-ounce cereal boxes of corn flakes. He found their mean weight to be $17.8$ ounces with a standard deviation of $0.4$ ounces. At $\alpha = 0.01$ , are the cereal boxes lighter than they should be?

State whether the null hypothesis should be rejected on the basis of the given $P$ -value. $P$ -value $= 0.001 , \alpha = 0.05$ , one-tailed test

A sample of 85 chewable vitamin tablets have a sample mean of 272 milligrams of vitamin C. Nutritionists want to perform a hypothesis test to determine how strong the Evidence is that the mean mass of vitamin C per tablet differs from 270 milligrams. State The appropriate null and alternate hypotheses.

The null hypothesis states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters.

When conducting a two-tailed $z$ test with $\alpha = 0.01$ , the test value was computed to be 2.07. The decision would be to not reject the null hypothesis.

A garbage collector believes that he averages picking up more than four tons of garbage per day. What is the null hypothesis for his statement?

A sample of 43 students enroll in a program that claims to improve scores on the quantitative reasoning portion of the Graduate Record Examination (GRE). The participants take a mock GRE test before the program begins and again at the end measure their improvement. The mean number of points improved wa $\bar { x } = 7$ . Assume the standard deviation is $\sigma = 56$ and let $\mu$ be the population mean number of points improved. To determine whether the program is effective, a test is made of the hypotheses $H _ { 0 } : \mu = 0$ versus $H _ { 1 } : \mu > 0$ . i). Compute the value of the test statistic. ii). Compute the $P$ -value. iii). Do you reject $H _ { 0 }$ at the $\alpha = 0.01$ level?

In a simple random sample of size 100 , there were 60 individuals in the category of interest. It is desired to test $H _ { 0 } : p = 0.69$ versus $H _ { 1 } : p < 0.69$ . i). Compute the sample proportion $\hat { p }$ . ii). It is desired to test $H _ { 0 } : p = 0.69$ versus $H _ { 1 } : p > 0.69$ Compute the test sti iii). Do you reject $H _ { 0 }$ at the $0.05$ level? tistic

Find the critical values for the following values of the significance level $\alpha$ , sample size $n$ , and alternate hypothesis $H _ { 1 }$ . $\alpha = 0.01 , n = 8 , H _ { 1 } : \mu \neq \mu _ { 0 }$

Thirty-eight 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 $\bar { x } = 16$ . Assume thi standard deviation is $\sigma = 67$ 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 $H _ { 1 } : \mu > 0$ . Do you reject $H _ { 0 }$ at the $\alpha = 0.05$ level?

A test of $H _ { 0 } : \mu = 43$ versus $H _ { 1 } : \mu \neq 43$ is performed using a significance level of $\alpha = 0.01$ . The $P$ -value is $0.129$ . Is $H _ { 0 }$ rejected?

A recent study of business travelers claims they spend an average of $\$ 41.00$ per day on meals. As a test of this claim, a random sampling of 16 business travelers found they had spent an average of $\$ 45.00$ per day with a standard deviation of $\$ 3.65$ . What are the critical values for a two-tailed $t$ test of this claim with $\alpha = 0.05$ ?

A scientist claims that only $65 \%$ of geese in his area fly south for the winter. He tags random geese in the summer and finds that 20 of them do not fly south in the winter. If $\alpha = 0.05$ , is the scientist's belief warranted?

In performing a hypothesis test, one should decide whether to reject or not reject the null hypothesis before summarizing the results.