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

I want to test H0: p = .7 vs. Ha: p ≠ .7 using a test of hypothesis. This test would be called a(n) ____________ test.

It is desired to test H0: μ = 45 against Ha: μ < 45 using α = .10. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If μ is really equal to 40, what is the probability that the hypothesis test would lead the investigator to commit a Type II error?

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 30 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 30 minutes. What assumption is necessary for this test to be valid?

A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 15% 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 60 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in 7. Calculate the test statistic used by the researchers for the corresponding test of hypothesis.

n = 700, p0 = 0.01

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 250 manufacturing firms is selected, and only 27 of them offer child-care benefits. Specify the rejection region that the union will use when testing at α = .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 sample of 8 measurements, randomly selected from a normally distributed population, resulted in the following summary statistics: $\bar { x }$ = 5.2, s = 1.1. Test the null hypothesis that the mean of the population is 4 against the alternative hypothesis μ ≠ 4. Use α = .05.

A random sample of n = 12 observations is selected from a normal population to test H0: μ = 22.1 against Ha: μ > 22.1 at α = .05. Specify the rejection region.

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 .07 to ensure proper inoculation. A random sample of 25 injections was measured. Suppose the p-value for the test is p = .0024. State the proper conclusion using α = .01.

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 0.02.

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 49 bottles and finds the average amount of liquid held by the bottles is 9.9155 ounces with a standard deviation of 0.35 ounce. Suppose the p-value of this test is 0.0455. State the proper conclusion.

Under the assumption that ? = ?a, where ?a is the alternative mean, the distribution of $x$ is mound shaped and symmetric about ?a.

It is desired to test H0: μ = 55 against Ha: μ < 55 using α = .10. The population in question is uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If μ is really equal to 50, what is the power of this test?

A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 20% 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 80 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in 9. Is the sample size sufficiently large to conduct this test of hypothesis? Explain.

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 are now 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. What null and alternative hypothesis should be tested? A) $\mathrm { H } _ { 0 } : \mu = 215$ vs. $\mathrm { H } _ { \mathrm { A } } : \mu \neq 215$ B) $\mathrm { H } _ { 0 } : \mu = 215$ vs. $\mathrm { H } _ { \mathrm { A } } : \mu < 215$ C) $\mathrm { H } _ { 0 } : \mu = 215$ vs. $\mathrm { H } _ { \mathrm { A } } : \mu > 215$ D) $\mathrm { H } _ { 0 } : \mu \geq 215$ vs. $\mathrm { H } _ { \mathrm { A } } : \mu < 215$ Answer the question True or False.

An ink cartridge for a laser printer is advertised to print an average of 10,000 businesses that have recently bought this cartridge are asked to report the number of 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 10,000 against the alternative hypothesis μ ≠ 10,000. Use α = .10.

According to an advertisement, a strain of soybeans planted on soil prepared with a specified fertilizer treatment has a mean yield of 594 bushels per acre. Twenty 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 20 farms are $\bar { x } = 550 \text { and } s ^ { 2 } = 10,000$ . Specify the null and alternative hypotheses used to determine if the mean yield for the soybeans is different than advertised.

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 μ0.

The smaller the p-value in a test of hypothesis, the more significant the results are.