Exam 8: Inferences Based on a Single

A small private college is interested in determining the percentage of its students who live off campus and drive to class. Specifically, it was desired to determine if less than 20% of their current Students live off campus and drive to class. Find the large-sample rejection region for the test of Interest to the college when using a level of significance of 0.01.

A national organization has been working with utilities throughout the nation to find sites for large wind machines that generate electricity. Wind speeds must average more than 11 miles per hour $( \mathrm { mph } )$ for a site to be acceptable. Recently, the organization conducted wind speed tests at a particular site. To determine whether the site meets the organization's requirements, consider the test, $H _ { 0 } : \mu = 11$ vs. $H _ { \mathrm { a } } : \mu > 11$ , where $\mu$ is the true mean wind speed at the site and $\alpha = .01$ . Suppose the observed significance level ( $p$ -value) of the test is calculated to be $p = 0.4052$ . Interpret this result.

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 $\alpha = .05$ to determine whether there is any evidence to support the counselor's suspicions.

A new apparatus has been devised to replace the needle in administering vaccines. The apparatus, which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the variance in the amount of serum injected to a given person must not be greater than .08 to ensure proper inoculation. A random sample of 25 injections resulted in a variance of .103. Specify the rejection region for the test. Use $\alpha = , 10$

The null distribution is the distribution of the test statistic assuming the null hypothesis is true; it mound shaped and symmetric about the null mean $\mu _ { 0 }$

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. $H _ { \mathrm { a } } : \sigma ^ { 2 } > 155 , n = 25 , \alpha = .10$

An ink cartridge for a laser printer is advertised to print an average of 10,000 pages. A random sample of eight businesses that have recently bought this cartridge are asked to report the number of pages printed by a single cartridge. The results are shown. 9771 9811 9885 9914 9975 10,079 10,145 10,214 Assume that the data belong to a normal population. Test the null hypothesis that the mean number of pages is 10,000 against the alternative hypothesis $\mu \neq 10,000$ . Use $\alpha = .10$ .

A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful only if the average time Spent on a delivery does not exceed 38 minutes. The owner has randomly selected 19 customers And delivered pizzas to their homes in order to test whether the mean delivery time actually Exceeds 38 minutes. Suppose the p-value for the test was found to be .0276. State the correct Conclusion.

Given $H _ { 0 } : \mu = 18 , H _ { \mathrm { a } } : \mu < 18$ , and $p = 0.068$ . Do you reject or fail to reject $H _ { 0 }$ at the $.05$ level of significance?

A national organization has been working with utilities throughout the nation to find sites for large wind machines that generate electricity. Wind speeds must average more than 25 miles per hour (mph) for a site to be acceptable. Recently, the organization conducted wind speed tests at a particular site. Based on a sample of $n = 43$ wind speed recordings (taken at random intervals), the wind speed at the site averaged $\bar { x } = 25.8 \mathrm { mph }$ , with a standard deviation of $s = 4.2 \mathrm { mph }$ . To determine whether the site meets the organization's requirements, consider the test, $H _ { 0 } : \mu = 25$ vs. $H _ { \mathrm { a } } : \mu > 25$ , where $\mu$ is the true mean wind speed at the site and $\alpha = .01$ . Suppose the value of the test statistic were computed to be $1.25$ . State the conclusion.

A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful only if the average time Spent on a delivery does not exceed 27 minutes. The owner has randomly selected 17 customers And delivered pizzas to their homes in order to test whether the mean delivery time actually Exceeds 27 minutes. What assumption is necessary for this test to be valid?

A new apparatus has been devised to replace the needle in administering vaccines. The apparatus, which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the variance in the amount of serum injected to a given person must not be greater than $.06$ to ensure proper inoculation. A random sample of 49 injections was measured. Suppose the $p$ -value for the test is $p = .0085$ . State the proper conclusion using $\alpha = .01$ .

A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful only if the average time Spent on a delivery does not exceed 38 minutes. The owner has randomly selected 27 customers And delivered pizzas to their homes. What hypotheses should the owner test to demonstrate that The pizza delivery will not be successful?

We have created a $99 \%$ confidence interval for $\mu$ with the result $( 8,13 )$ . What conclusion will we make if we test $H _ { 0 } : \mu = 15$ vs. $H _ { \mathrm { a } } : \mu \neq 15$ at $\alpha = .01$ ?

Researchers have claimed that the average number of headaches per student during a semester Statistics is 14. Statistics students believe the average is higher. In a sample of $n = 15$ students th mean is 19 headaches with a deviation of 1.9. Which of the following represent the null and alternative hypotheses necessary to test the students' belief?

A consumer product magazine recently ran a story concerning the increasing prices of digital cameras. The story stated that digital camera prices dipped a couple of years ago, but now are Beginning to increase in price because of added features. According to the story, the average price Of all digital cameras a couple of years ago was $215.00. A random sample of cameras was recently Taken and entered into a spreadsheet. It was desired to test to determine if that average price of all Digital cameras is now more than $215.00. The information was entered into a spreadsheet and the Following printout was obtained: One-Sample T Test Null Hypothesis: $\mu = 215$ Alternative Hyp: $\mu > 215$ Cases Included 22 Use the p- value given above to determine which of the following conclusions is correct.

The hypotheses for $H _ { 0 } : \mu = 65$ and $H _ { \mathrm { a } } : \mu > 65$ are tested at $\alpha = .05$ . Sketch the appropriate rejection region.

According to an advertisement, a strain of soybeans planted on soil prepared with a specified fertilizer treatment has a mean yield of 99 bushels per acre. Fifteen farmers who belong to a cooperative plant the soybeans in soil prepared as specified. Each uses a 40 -acre plot and records the mean yield per acre. The mean and variance for the sample of the 15 farms are $\bar { x } = 84$ and $s ^ { 2 } = 10,125$ . Find the rejection region used for determining if the mean yield for the soybeans is not equal to 99 bushels per acre. Use $\alpha = .05$ .

For the given rejection region, sketch the sampling distribution for z and indicate the location of the rejection region. -z < -2.33 or z > 2.33

A bottling company produces bottles that hold 10 ounces of liquid. Periodically, the company gets complaints that their bottles are not holding enough liquid. To test this claim, the bottling company randomly samples 22 bottles and finds the average amount of liquid held by the bottles is $9.7$ ounces with a standard deviation of $.2$ ounce. Which of the following is the set of hypotheses the company wishes to test?