Exam 8: Inferences Based on a Single

The rejection region refers to the values of the test statistic for which we will reject the alternative hypothesis.

State University uses thousands of fluorescent light bulbs each year. The brand of bulb it currently uses has a mean life of 600 hours. A competitor claims that its bulbs, which cost the same as the brand the university currently uses, have a mean life of more than 600 hours. The university has decided to purchase the new brand if, when tested, the evidence supports the manufacturer's claim at the $.05$ significance level. Suppose 100 bulbs were tested with the following results: $\bar { x } = 627.5$ hours, $s = 100$ hours. Conduct the test using $\alpha = .05$ .

Consider a test of $\mathrm { H } _ { 0 } : \mu = 4$ . For the following case, give the rejection region for the test in terms of the $z$ -statistic: $\mathrm { H } _ { \mathrm { a } } : \mu \neq 4 , \alpha = 0.01$

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 11 ounces printed on each cartridge. To check this claim, a sample of $n = 23$ cartridges are randomly selected from the shipment and carefully weighed. Summary statistics for the sample are: $\bar { x } = 11.14$ ounces, $s = .23$ ounce. To determine whether the supplier's claim is true, consider the test, $H _ { 0 } : \mu = 11$ vs. $H _ { \mathrm { a } } : \mu > 11$ , where $\mu$ is the true mean weight of the cartridges. Calculate the value of the test statistic.

A company reports that 80% of its employees participate in the companyʹs stock purchase plan. A random sample of 50 employees was asked the question, ʺDo you participate in the stock purchase plan?ʺ The answers are shown below. yes no no yes no no yes yes no no no yes yes yes no yes no no yes yes no yes yes no yes yes no yes yes yes yes no no yes yes yes yes yes no yes no yes yes no yes yes yes yes yes yes Perform the appropriate test of hypothesis to investigate your suspicion that fewer than $80 \%$ of the company's employees participate in the plan. Use $\alpha = .05$ .

The scores on a standardized test are reported by the testing agency to have a mean of 75. 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 Find and interpret the p -value for the test of :\mu=75 against :\mu>75 .

A random sample of $n = 12$ observations is selected from a normal population to test $H _ { 0 } : \mu$ $= 22.1$ against $H _ { \mathrm { a } } : \mu > 22.1$ at $\alpha = .05$ . Specify the rejection region.

Consider a test of $\mathrm { H } _ { 0 } : \mu = 8$ . For the following case, give the rejection region for the test in terms of the $z$ -statistic: $\mathrm { H } _ { \mathrm { a } } : \mu > 8 , \alpha = 0.10$

A significance level for a hypothesis test is given as $\alpha = .01$ . Interpret this value.

If a hypothesis test were conducted using $\alpha = 0.05$ , to which of the following $p$ -values would cause the null hypothesis to be rejected.

A company claims that 9 out of 10 doctors (i.e., $90 \%$ ) recommend its brand of cough syrup to their patients. To test this claim against the alternative that the actual proportion is less than $90 \%$ , a random sample of doctors was taken. Suppose the test statistic is $z = - 1.75$ . Can we conclude that $H _ { 0 }$ should be rejected at the a) $\alpha = .10$ , b) $\alpha = .05$ , and c) $\alpha = .01$ level?

It has been estimated that the $\mathrm { G }$ -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 $64 \mathrm { G }$ -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, $s = 8$ miles per gallon. Calculate the power of the test if the value of the mean is 31 miles per gallon. Use a value of $\alpha = .025$ .

Data were collected from the sale of 25 properties by a local real estate agent. The following printout concentrated on the land value variable from the sampled properties. HYPOTHESIS: MEAN X $= \mathrm { x }$ Find the $p$ -value for testing whether the mean land value differs from $\$ 47,768$ .

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 49 injections resulted in a variance of .118. Calculate the test statistic for the test of interest.

A company claims that 9 out of 10 doctors (i.e., $90 \%$ ) recommend its brand of cough syrup to their patients. To test this claim against the alternative that the actual proportion is less than $90 \%$ , a random sample of 100 doctors was chosen which resulted in 87 who indicate that they recommend this cough syrup. The test statistic in this problem is approximately:

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 12 ounces printed on each cartridge. To check this claim, a sample of $n = 15$ cartridges are randomly selected from the shipment and carefully weighed. Summary statistics for the sample are: $\bar { x } = 12.13$ ounces, $s = .30$ ounce. To determine whether the supplier's claim is true, consider the test, $H _ { 0 } : \mu " 12$ vs. $H _ { \mathrm { a } } : \mu > 12$ , where $\mu$ is the true mean weight of the cartridges. Find the rejection region for the test using $\alpha = .01$ .

Consider the following printout. HYPOTHESIS: VARIANCE $X = x$ $X = \text { gpa }$ SAMPLE MEAN OF $\mathrm { X } = 2.8814$ SAMPLE VARIANCE OF $X = .21000$ SAMPLE SIZE OF $X = 250$ HYPOTHESIZED VALUE $( \mathrm { x } ) = 3.0$ VARIANCE $\mathrm { X } - \mathrm { x } = - .1186$ $z = - 4.09209$ Is this a large enough sample for this analysis to work?

In a test of $H _ { 0 } : \mu = 70$ against $H _ { \mathrm { a } } : \mu \neq 70$ , the sample data yielded the test statistic $z = 2.11$ . Find and interpret the $p$ -value for the test.

The State Association of Retired Teachers has recently taken flak from some of its members regarding the poor choice of the associationʹs name. The associationʹs by-laws require that more Than 60 percent of the association must approve a name change. Rather than convene a meeting, it Is first desired to use a sample to determine if meeting is necessary. Suppose the association Decided to conduct a test of hypothesis using the following null and alternative hypotheses: :=0.6 :>0.6 Define a Type II Error in the context of this problem.

For the given rejection region, sketch the sampling distribution for z and indicate the location of the rejection region. -z > 2.575