Exam 7: Inferences Based on a Single Sample: 355 Tests of Hypotheses

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. What assumptions are necessary for any inferences derived from this printout to be valid?

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 0.01.

For the given rejection region, sketch the sampling distribution for z and indicate the location of the rejection region. - $z < - 2.33 \text { or } z > 2.33$

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

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

Researchers have claimed that the average number of headaches per student during a semester of Statistics is 11. Statistics students believe the average is higher. In a sample of $n = 16$ students the mean is 12 headaches with a deviation of 2.4. Which of the following represent the null and alternative hypotheses necessary to test the students' belief?

I want to test $H _ { 0 } : p = .4$ vs. $H _ { \mathrm { a } } : p \neq .4$ using a test of hypothesis. If I concluded that $p$ is $.4$ when, in fact, the true value of $p$ is not $.4$ , then I have made a______

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 10,000 people yielded the following data on the number of tissues used during a cold: $\bar { x } = 56 , s = 25$ . We want to test the alternative hypothesis $H _ { \mathrm { a } } : \mu < 62$ . State the correct rejection region for $\alpha = .05$ .

A supermarket sells rotisserie chicken at a fixed price per chicken rather than by the weight of the chicken. The store advertises that the average weight of their chickens is 4.6 pounds. A random sample of 30 of the store's chickens yielded the weights (in pounds) shown below. 4.4 4.7 4.6 4.4 4.5 4.3 4.6 4.5 4.6 4.9 4.6 4.8 4.3 4.4 4.7 4.5 4.2 4.3 4.1 4.0 4.5 4.6 4.2 4.4 4.7 4.8 5.0 4.2 4.1 4.5 Find and interpret the $p$ -value in a test of $H _ { 0 } : \mu = 4.6$ against $H _ { a } : \mu < 4.6$ .

The null hypothesis represents the status quo to the party performing the sampling experiment.

Consider a test of $\mathrm { H } _ { 0 } : \mu = 85$ performed with the computer. SPSS reports a two-tailed p-value of 0.0070. Make the appropriate conclusion for the given situation: $\mathrm { H } _ { \mathrm { a } } : \mu \neq 85 , z = 2.7 , \alpha = 0.01$

Consider a test of $\mathrm { H } _ { 0 } : \mu = 60$ performed with the computer. SPSS reports a two-tailed p-value of 0.0892. Make the appropriate conclusion for the given situation: $\mathrm { H } _ { \mathrm { a } } : \mu < 60 , z = - 1.7 , \alpha = 0.05$

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 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 70 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in 8. Calculate the test statistic used by the researchers for the corresponding test of hypothesis.

For the given value of α and observed significance level (p-value), indicate whether the null hypothesis would be rejected. - $\alpha = 0.01 , \mathrm { p } \text {-value } = 0.05$

State University uses thousands of fluorescent light bulbs each year. The brand of bulb it currently uses has a mean life of 760 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 760 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 92 bulbs were tested with the following results: $\bar { x } = 778$ hours, $s = 91$ hours. Find the rejection region for the test of interest to the State University.

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 company reports that 80% of its employees participate in the company's stock purchase plan. A random sample of 50 employees was asked the question, "Do you participate in the stock purchase plan?" 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 .

For the given value of ? and observed significance level (p-value), indicate whether the null hypothesis would be rejected. - $\alpha = 0.1 , \mathrm { p } \text {-value } = 0.04$