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

A random sample of $n = 18$ observations is selected from a normal population to test $H _ { 0 }$ $= 145$ against $H _ { \mathrm { a } } : \mu \neq 145$ at $\alpha = .10$ . Specify the rejection region.

Based on the information in the screen below, what would you conclude in the test of $H _ { 0 } : \mu \leq 14 , H _ { \mathrm { a } } : \mu > 14$ . Use $\alpha = .01$ .

For the given rejection region, sketch the sampling distribution for z and indicate the location of the rejection region. - $z < - 1.96 \text { or } z > 1.96$

An industrial supplier has shipped a truckload of teflon lubricant cartridges to an aerospace customer. The customer has been assured that the mean weight of these cartridges is in excess of the 13 ounces printed on each cartridge. To check this claim, a sample of $n = 21$ cartridges are randomly selected from the shipment and carefully weighed. Summary statistics for the sample are: $\bar { x } = 13.11$ ounces, $s = .21$ ounce. To determine whether the supplier's claim is true, consider the test, $H _ { 0 } : \mu = 13$ vs. $H _ { \mathrm { a } } : \mu > 13$ , where $\mu$ is the true mean weight of the cartridges. Calculate the value of the test statistic.

An insurance company sets up a statistical test with a null hypothesis that the average time for processing a claim is 7 days, and an alternative hypothesis that the average time for processing a claim is greater than 7 days. After completing the statistical test, it is concluded that the average time exceeds 7 days. However, it is eventually learned that the mean process time is really 7 days. What type of error occurred in the statistical test?

The business college computing center wants to determine the proportion of business students who have laptop computers. If the proportion exceeds 35%, then the lab will scale back a proposed enlargement of its facilities. Suppose 300 business students were randomly sampled and 75 have laptops. What assumptions are necessary for this test to be satisfied?

If I specify $\beta$ to be .24, then the value of α must be .76.

Consider the following printout. HYPOTHESIS: VARIANCE $X = x$ $X = \text { gpa }$ SAMPLE MEAN OF $X = 2.5516$ SAMPLE VARIANCE OF X $= .24000$ SAMPLE SIZE OF $X = 157$ HYPOTHESIZED VALUE $( x ) = 2.7$ VARIANCE $X - x = - .1484$ $z = - 3.79558$ Is this a large enough sample for this analysis to work?

How many tissues should a package of tissues contain? Researchers have determined that a person uses an average of 61 tissues during a cold. Suppose a random sample of 2500 people yielded the following data on the number of tissues used during a cold: $\bar { x } = 51 , s = 25$ . Suppose the corresponding test statistic falls in the rejection region at $\alpha = .05$ . What is the correct conclusion?

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 .4 ounce. Calculate the appropriate test statistic.

Consider a test of $\mathrm { H } _ { 0 } : \mu = 85$ performed with the computer. SPSS reports a two-tailed p-value of $0.2714$ . Make the appropriate conclusion for the given situation: $\mathrm { H } _ { \mathrm { a } } : \mu > 85 , z = 1.10 , \alpha = 0.10$

A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 51 students and asked each to provide the amount of time they spent traveling to campus. The sample results found that the sample mean was $23.243$ minutes and the sample standard deviation was $20.40$ minutes. Find the rejection region for determining if the population standard deviation exceeds 20 minutes. Use $\alpha = 0.05$ .

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 $n = 200$ 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 $\$ 215.00$ . Find the large-sample rejection region appropriate for this test if we are using $\alpha = 0.05$ .

Find the rejection region for the specified hypothesis test. -Consider a test of $\mathrm { H } _ { 0 } : \mu = 9$ . For the following case, give the rejection region for the test in terms of the $z$ -statistic: $\mathrm { H } _ { \mathrm { a } } : \mu < 9 , \alpha = 0.01$

The rejection region for a two-tailed test with $\alpha = .05 \text { is } - 1.96 < z < 1.96$

What is the probability associated with not making a Type II error?

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 .05 to ensure proper inoculation. A random sample of 25 injections resulted in a variance of .118. Specify the rejection region for the test. Use $\alpha = 10$

It has been estimated that the $G$ -car obtains a mean of 30 miles per gallon on the highway, and the company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects $49 \mathrm { G }$ -cars and records the mileage obtained for each car over a driving course similar to that used to obtain the estimate. The following data resulted: $\bar { x } = 31.2$ miles per gallon, $s = 7$ miles per gallon. Calculate the value of $\beta$ if the true value of the mean is 32 miles per gallon. Use $\alpha = .025$ .

Increasing numbers of businesses are offering child-care benefits for their workers. However, one union claims that more than 90% of firms in the manufacturing sector still do not offer any child-care benefits. A random sample of 250 manufacturing firms is selected and asked if they offer child-care benefits. Suppose the p-value for this test was reported to be $p = .1103$ State the conclusion of interest to the union. $\text { Use } \alpha = .10$

Identify the observed level of significance for the test summarized on the screen below and interpret its value.