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

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. $H _ { \mathrm { a } } : \sigma ^ { 2 } \approx 155 , n = 10 , \alpha = .05$ A) $\chi ^ { 2 } < 2.70039$ or $\chi ^ { 2 } > 19.0228$ B) $2.70039 < \chi ^ { 2 } < 19.0228$ C) $\chi ^ { 2 } < 3.32511$ or $\chi ^ { 2 } > 16.9190$ D) $\chi ^ { 2 } < 3.24697$ or $\chi ^ { 2 } > 20.4831$

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. Find the large-sample rejection region for the test of interest to the college when using a level of significance of 0.05.

Suppose we wish to test $H _ { 0 } : \mu = 58 \text { vs. } H _ { \mathrm { a } } : \mu > 58 \text {. }$ . What will result if we conclude that the mean is greater than 58 when its true value is really 63?

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) fail to reject $H _ { 0 }$ B) reject $H _ { 0 }$ C) not sufficient information to decide

An insurance company sets up a statistical test with a null hypothesis that the average time for processing a claim is 4 days, and an alternative hypothesis that the average time for processing a claim is greater than 4 days. After completing the statistical test, it is concluded that the average time exceeds 4 days. However, it is eventually learned that the mean process time is really 4 days. What type of error occurred in the statistical test?

A bottling company produces bottles that hold 8 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 24 bottles and finds the average amount of liquid held by the bottles is 7.6 ounces with a standard deviation of .3 ounce. Calculate the appropriate test statistic.

For the given rejection region, sketch the sampling distribution for z and indicate the location of the rejection region. -z < -1.28

n = 100, p0 = 0.4

The value of $\beta$ is the area under the bell curve for the distribution of $\bar { x }$ centered at $\mu _ { \mathrm { a } }$ for values of $\bar { x }$ that fall within the acceptance region of the distribution of $\bar { x }$ centered at $\mu _ { 0 }$ .

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 Is a sample size n = 22 large enough to utilize the central limit theorem in this inferential procedure?

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 10 miles per hour (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, H0: μ = 10 vs. Ha: μ > 10, where μ is the true mean wind speed at the site and α = .01. Suppose the observed significance level (p-value) of the test is calculated to be p = 0.2991. Interpret this result.

An educational testing service designed an achievement test so that the range in student scores would be greater than 420 points. To determine whether the objective was achieved, the testing service gave the test to a random sample of 30 students and found that the sample mean and variance were 759 and 1943, respectively. Conduct the test for $H _ { 0 } : \sigma ^ { 2 } = 4900 \text { vs. } H _ { a } : \sigma ^ { 2 } > 4900 \text { using } \alpha = .025 \text {. Assume the range is } 6 \sigma \text {. }$

How many tissues should a package of tissues contain? Researchers have determined that a person uses an average of 50 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: $\bar { x } = 45 , s = 18$ . Using the sample information provided, set up the calculation for the test statistic for the relevant hypothesis test, but do not simplify. A) $z = \frac { 45 - 50 } { \frac { 18 } { \sqrt { 10,000 } } }$ B) $z = \frac { 45 - 50 } { 18 }$ C) $z=\frac{45-50}{\frac{18}{10,000^{2}}}$ D) $z = \frac { 45 - 50 } { \frac { 18 ^ { 2 } } { 10,000 } }$

$\alpha = 0.08 , \mathrm { p } \text {-value } = 0.10$ A) Fail to reject $\mathrm { H } _ { 0 }$ B) Reject $\mathrm { H } _ { 0 }$

We have created a $99 \%$ confidence interval for $\mu$ with the result $( 10,15 )$ . What conclusion will we make if we test $H _ { 0 } : \mu = 17$ vs. $H _ { \mathrm { a } } : \mu \neq 17$ at $\alpha = .01$ ? A) Reject $\mathrm { H } _ { 0 }$ in favor of $H _ { a }$ . B) Accept $\mathrm { H } _ { 0 }$ rather than $\mathrm { H } _ { \mathrm { a } }$ - C) Fail to reject $\mathrm { H } _ { 0 }$ . D) We cannot tell what our decision will be with the information given.

Consider a test of H0: μ = 65 performed with the computer. SPSS reports a two-tailed p-value of 0.0892. Make the appropriate conclusion for the given situation: Ha: μ < 65, z = -1.7, α = 0.05

Increasing numbers of businesses are offering child-care benefits for their workers. However, one union claims that more than 85% of firms in the manufacturing sector still do not offer any child-care benefits. A random sample of 320 manufacturing firms is selected and asked if they offer child-care benefits. Suppose the p-value for this test was reported to be p = .1124. State the conclusion of interest to the union. Use α = .10.

A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 17% of women who actually have the disease. A new method has been developed that researchers hope will be able to detect cancer more accurately. A random sample of 74 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in ten. Specify the null and alternative hypotheses that the researchers wish to test.

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. The college decided to take a random sample of 108 of their current students to use in the analysis. Is the sample size of n = 108 large enough to use this inferential procedure?

How many tissues should a package of tissues contain? Researchers have determined that a person uses an average of 62 tissues during a cold. Suppose a random sample of 100 people yielded the following data on the number of tissues used during a cold: $\bar { x }$ = 50, s = 16. Identify the null and alternative hypothesis for a test to determine if the mean number of tissues used during a cold is less than 62. A) $H _ { 0 } : \mu = 62$ vs. $H _ { a } : \mu < 62$ B) $H _ { 0 } : \mu = 62$ vs. $H _ { \mathrm { a } } : \mu > 62$ C) $H _ { 0 } : \mu = 62$ vs. $H _ { \mathrm { a } } : \mu \neq 62$ D) $H _ { 0 } : \mu > 62$ vs. $H _ { \mathrm { a } } : \mu \leq 62$