Exam 8: Inferences Based on a Single Sample: Tests of Hypothesis

State University uses thousands of fluorescent light bulbs each year. The brand of bulb it currently uses has a mean life of 940 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 940 hours. The university has decided to purchase the new brand if, when tested, the evidence supports the manufacturer's claim at the .01 significance level. Suppose 99 bulbs were tested with the following results: $\bar { x }$ = 962 hours, s = 77 hours. Find the rejection region for the test of interest to the State University.

A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 238 students and asked each to provide the amount of time they spent traveling to campus. This variable, travel time, was then used conduct a test of hypothesis. The goal was to determine if the average travel time of all the university's students differed from 20 minutes. Suppose the sample mean and sample standard deviation were calculated to be 23.2 and 20.26 minutes, respectively. Calculate the value of the test statistic to be used in the test.

Given $H _ { 0 } : \mu = 25 , H _ { \mathrm { a } } : \mu \neq 25$ , and $p = 0.034$ . Do you reject or fail to reject $H _ { 0 }$ at the $.01$ level of significance? A) fail to reject $H _ { 0 }$ B) reject $\mathrm { H } _ { 0 }$ C) not sufficient information to decide

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. Calculate the test statistic for the test of interest.

Consider a test of H0: μ = 30 performed with the computer. SPSS reports a two-tailed p-value of 0.0164. Make the appropriate conclusion for the given situation: Ha: μ > 30, z = -2.4, α = 0.01

If I specify β to be .36, then the value of α must be .64.

A revenue department is under orders to reduce the time small business owners spend filling out pension form ABC-5500. Previously the average time spent on the form was 6.3 hours. In order to test whether the time to fill out the form has been reduced, a sample of 53 small business owners who annually complete the form was randomly chosen, and their completion times recorded. The mean completion time for ABC-5500 form was 6.1 hours with a standard deviation of 2.6 hours. In order to test that the time to complete the form has been reduced, state the appropriate null and alternative hypotheses. A) $\mathrm { H } _ { 0 } : \mu = 6.3$ $H _ { \mathrm { a } } : \mu < 6.3$ B) $H _ { 0 } : \mu = 6.3$ $H _ { \mathrm { a } } : \mu > 6.3$ C) $H _ { 0 } : \mu = 6.3$ $H_{\mathrm{a}}: \mu \neq 6.3$ D) $H _ { 0 } : \mu > 6.3$ $H _ { \mathrm { a } } : \mu < 6.3$

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 25 bottles and finds the average amount of liquid held by the bottles is 9.8 ounces with a standard deviation of .4 ounce. Which of the following is the set of hypotheses the company wishes to test?

How many tissues should a package of tissues contain? Researchers have determined that a person uses an average of 66 tissues during a cold. Suppose a random sample of 10,000 people yielded the following data on the number of tissues used during a cold: $\bar { x } = 61 , s = 25$ . We want to test the alternative hypothesis $H _ { \mathrm { a } } : \mu < 66$ . State the correct rejection region for $\alpha = .05$ . A) Reject $H _ { 0 }$ if $z < - 1.645$ . B) Reject $H _ { 0 }$ if $z > 1.645$ . C) Reject $H _ { 0 }$ if $z > 1.96$ or $z < - 1.96$ . D) Reject $H _ { 0 }$ if $z < - 1.96$ .

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 14 ounces printed on each cartridge. To check this claim, a sample of n = 17 cartridges are randomly selected from the shipment and carefully weighed. Summary statistics for the sample are: $\bar { x }$ = 14.1 ounces, s = .17 ounce. To determine whether the supplier's claim is true, consider the test, H0: ? = 14 vs. Ha: ? > 14, where ? is the true mean weight of the cartridges. Calculate the value of the test statistic.

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 H0: μ = 75 against Ha: μ > 75.

How many tissues should a package of tissues contain? Researchers have determined that a person uses an average of 69 tissues during a cold. Suppose a random sample of 2500 people yielded the following data on the number of tissues used during a cold: $\bar { x } = 60 , s = 16$ . Suppose the corresponding test statistic falls in the rejection region at $\alpha = , 05$ . What is the correct conclusion? A) At $\alpha = .05$ , reject $H _ { 0 }$ . B) At $\alpha = .10$ , reject $H _ { 0 }$ . C) At $\alpha = .05$ , accept $H _ { a }$ . D) At $\alpha = .10$ , reject $H _ { a }$ .

A revenue department is under orders to reduce the time small business owners spend filling out pension form ABC-5500. Previously the average time spent on the form was 67 hours. In order to test whether the time to fill out the form has been reduced, a sample of 81 small business owners who annually complete the form was randomly chosen and their completion times recorded. The mean completion time for the sample was 66.7 hours with a standard deviation of 15 hours. State the rejection region for the desired test at α = .10.

Consider a test of $\mathrm { H } _ { 0 } : \mu = 7$ . For the following case, give the rejection region for the test in terms of the $z$ -statistic: $\mathrm { H } _ { \mathrm { a } } : \mu \neq 7 , \alpha = 0.01$ A) $| z | > 2.575$ B) $z > 2.575$ C) $z > 2.33$ D) $| z | > 2.33$

In a test of H0: μ = 70 against Ha: μ ≠70, the sample data yielded the test statistic z = 2.11. Find and interpret the p-value for the test.

Consider a test of H0: μ = 70 performed with the computer. SPSS reports a two-tailed p-value of 0.2302. Make the appropriate conclusion for the given situation: Ha: μ > 70, z = 1.20, α = 0.10

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

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 83 who indicate that they recommend this cough syrup. The test statistic in this problem is approximately:

Consider the following printout. HYPOTHESIS: VARIANCE $\mathrm { X } = \mathrm { x }$ $X = \text { gpa }$ SAMPLE MEAN OF X =2.2911 SAMPLE VARIANCE OF X =.18000 SAMPLE SIZE OF X =199 HYPOTHESIZED VALUE (x)=2.4 VARIANCE X-x =-.1089 z =-3.62091 Suppose we tested Ha: ? < 2.4. Find the appropriate rejection region if we used ? = .05.

A __________ is a numerical quantity computed from the data of a sample and is used in reaching a decision on whether or not to reject the null hypothesis.