Exam 8: Inferences Based on a Single

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. A sample of 108 students was randomly selected and The following printout was obtained: Hypothesis Test - One Proportion Based on the information contained in the printout, what conclusion would be correct when testing at $\alpha = 0.05$ .

Solve the problem. -The rejection region for a two-tailed test with α = .05 is -1.96 < z < 1.96.

The business college computing center wants to determine the proportion of business students who have laptop computers. If the proportion exceeds $25 \%$ , then the lab will scale back a proposed enlargement of its facilities. Suppose 250 business students were randomly sampled and 75 have laptops. What assumptions are necessary for this test to be satisfied?

I want to test $H _ { 0 } : p = .7 \mathrm { vs. } H _ { \mathrm { a } } : p \neq .7$ using a test of hypothesis. If I concluded that $p$ is $.7$ when, in fact, the true value of $p$ is not $.7$ , then I have made a________

Solve the problem. -Consider the following printout. HYPOTHESIS: VARIANCE X $= \mathrm { x }$ $X = \text { gpa }$ SAMPLE MEAN OF X $= 2.6919$ SAMPLE VARIANCE OF $X = .19000$ SAMPLE SIZE OF $X = 217$ HYPOTHESIZED VALUE $( \mathrm { x } ) = 2.8$ VARIANCE X-x =-.1081 z =-3.65324 Suppose we tested $H _ { \mathrm { a } } : \mu < 2.8$ . Find the appropriate rejection region if we used $\alpha = .05$ .

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 420 manufacturing firms is selected and asked if they offer child-care benefits. Suppose the $p$ -value for this test was reported to be $p = .1066$ . State the conclusion of interest to the union. Use $\alpha = .10$ .

For the given binomial sample size and null-hypothesized value of $\mathrm { P } 0 ,$ determine whether the sample size is large enough to use the normal approximation methodology to conduct a test of the null hypothesis $\mathrm { H } _ { 0 } : \mathrm { p } " \mathrm { P } 0$ - $\mathrm { n } = 600 , \mathrm { p } 0 = 0.02$

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 tha the mean of the population is 4 against the alternative hypothesis $\mu \neq 4$ . Use $\alpha = .05$ .

According to an advertisement, a strain of soybeans planted on soil prepared with a specified fertilizer treatment has a mean yield of 528 bushels per acre. Twenty-five 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 25 farms are $\bar { x } = 484$ and $s ^ { 2 } = 9170$ . Specify the null and alternative hypotheses used to determine if the mean yield for the soybeans is different than advertised.

A Type I error occurs when we accept a false null hypothesis.

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?

Solve the problem. -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 10,000 people yielded the following data on the number of tissues used during a cold: $\bar { x } = 47 , s = 20$ . We want to test the alternative hypothesis $H _ { \mathrm { a } } : \mu < 61$ . State the correct rejection region for $\alpha = .05$ .

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

Consider a test of $\mathrm { H } _ { 0 } : \mu = 35$ performed with the computer. SPSS reports a two-tailed p-value of 0.0124. Make the appropriate conclusion for the given situation: $\mathrm { H } _ { \mathrm { a } } : \mu > 35 , z = - 2.5 , \alpha = 0.01$

A random sample of 100 observations is selected from a binomial population with unknown probability of success, $p$ . The computed value of $\hat { p }$ is equal to $.56$ . Find the observed levels of significance in a test of $H _ { 0 } : p = .5$ against $H _ { \mathrm { a } } : p > .5$ . Interpret the result.

A bottling company produces bottles that hold 12 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 64 bottles and finds the average amount of liquid held by the bottles Is 11.9155 ounces with a standard deviation of 0.40 ounce. Suppose the p-value of this test is 0.0455. State the proper conclusion.

I want to test $H _ { 0 } : p = .7 \mathrm { vs } . H _ { \mathrm { a } } : p \neq .7$ using a test of hypothesis. This test would be called a(n) ________test.

Given $H _ { 0 } : \mu = 25 , H _ { \mathrm { a } } : \mu \neq 25$ , and $p = 0.033$ . Do you reject or fail to reject $H _ { 0 }$ at the $.01$ level of significance?

Under the assumption that $\mu = \mu _ { \mathrm { a } }$ , where $\mu _ { \mathrm { a } }$ is the alternative mean, the distribution of $\bar { x }$ is mound shaped and symmetric about $\mu _ { \mathrm { a } }$ .