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

The rejection region refers to the values of the test statistic for which we will reject the alternative hypothesis.

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. Suppose a sample of 108 students produced a test statistic of z = -1.35. Find the p-value for the test of interest to the college.

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. HYPOTHESIS: MEAN X = x X = land_value SAMPLE MEAN OF X = 50,098 SAMPLE VARIANCE OF X = 273,643,254 SAMPLE SIZE OF X = 25 x = 45,655 MEAN X - x = 4443 t = 1.34293 D.F. = 24 P-VALUE = 0.1918585 P-VALUE/2 = 0.0959288 SD. ERROR = 3308.43 Find the p-value for testing whether the mean land value differs from $45,655.

We do not accept H0 because we are concerned with making a Type II error.

The scores on a standardized test are reported by the testing agency to have a mean of 70. 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 Use the data to conduct a test of hypotheses at α = .05 to determine whether there is any evidence to support the counselor's suspicions.

The hypotheses for $H _ { 0 } : \mu = 125.4 \text { and } H _ { \mathrm { a } } : \mu \neq 125.4 \text { are tested at } \alpha = .10 \text {. }$ . Sketch the appropriate rejection region.

It has been estimated that the G-car obtains a mean of 30 miles per gallon on the highway, and the company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects 36 G-cars and records the mileage obtained for each car over a driving course similar to that used to obtain the estimate. The following data resulted: x = 31.8 miles per gallon, s = 6 miles per gallon. Calculate the power of the test if the true value of the mean is 31 miles per gallon. Use a value of α = .025.

According to an advertisement, a strain of soybeans planted on soil prepared with a specified fertilizer treatment has a mean yield of 107 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 the 20 farms are $\bar { x } = 92 \text { and } s ^ { 2 } = 18,000$ . Find the rejection region used for determining if the mean yield for the soybeans is not equal to 107 bushels per acre. Use α = .05.

The State Association of Retired Teachers has recently taken flak from some of its members regarding the poor choice of the association's name. The association's by-laws require that more than 60 percent of the association must approve a name change. Rather than convene a meeting, it is first desired to use a sample to determine if meeting is necessary. Suppose the association decided to conduct a test of hypothesis using the following null and alternative hypotheses: H0: p = 0.6 HA: p > 0.6 Define a Type II Error in the context of this problem.

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. HYPOTHESIS: MEAN X = x X = land_value SAMPLE MEAN OF X = 51,860 SAMPLE VARIANCE OF X = 273,643,254 SAMPLE SIZE OF X = 25 x = 47,417 MEAN X - x = 4443 t = 1.34293 D.F. = 24 P-VALUE = 0.1918585 P-VALUE/2 = 0.0959288 SD. ERROR = 3308.43 What assumptions are necessary for any inferences derived from this printout to be valid?

State University uses thousands of fluorescent light bulbs each year. The brand of bulb it currently uses has a mean life of 800 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 800 hours. The university has decided to purchase the new brand if, when tested, the evidence supports the manufacturer's claim at the .05 significance level. Suppose 121 bulbs were tested with the following results: $\bar { x }$ = 827.5 hours, s = 110 hours. Conduct the test using α = .05.

The State Association of Retired Teachers has recently taken flak from some of its members regarding the poor choice of the association's name. The association's by-laws require that more than 60 percent of the association must approve a name change. Rather than convene a meeting, it is first desired to use a sample to determine if meeting is necessary. Identify the null and alternative hypothesis that should be tested to determine if a name change is warranted. A) $\mathrm { H } _ { 0 } : \mathrm { p } = 0.6$ vs. $\mathrm { H } _ { \mathrm { a } } : \mathrm { p } \neq 0.6$ B) $\mathrm { H } _ { 0 } : \mathrm { p } = 0.6$ vs. $\mathrm { H } _ { \mathrm { a } } : \mathrm { p } > 0.6$ C) $\mathrm { H } _ { 0 } : \mathrm { p } = 0.6$ vs. $\mathrm { H } _ { \mathrm { a } } : \mathrm { p } < 0.6$ D) $\mathrm { H } _ { 0 } : \mathrm { p } \geq 0.6$ vs. $\mathrm { H } _ { \mathrm { a } } : \mathrm { p } < 0.6$

In a test of H0: μ = 65 against Ha: μ > 65, the sample data yielded the test statistic z = 1.38. Find and interpret the p-value for the test.

n = 1100, p0 = 0.99

A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 51 students and asked each to provide the amount of time they spent traveling to campus. The sample results found that the sample mean was 23.243 minutes and the sample standard deviation was 20.40 minutes. It is desired to determine if the population standard deviation exceeds 20 minutes. Calculate the test statistic for this test of hypothesis. A) $\chi ^ { 2 } = 51$ B) $\chi ^ { 2 } = 53.06$ C) $\chi ^ { 2 } = 52.02$ D) $\chi ^ { 2 } = 58.11$

I want to test $H _ { 0 } : p = .7 \text { vs. } H _ { \mathrm { a } } : p \neq .7$ 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 __________.

A test of hypothesis was performed to determine if the true proportion of college students who preferred a particular brand of soda differs from .50. The ASP printout is supplied below. Note: All data refer to the proportion of students who preferred the brand of soda. HYPOTHESIS: PROPORTION X = x X = drink_(soda=1) SAMPLE PROPORTION OF X = .419162 SAMPLE SIZE OF X = 167 HYPOTHESIZED VALUE (x) = .5 SAMPLE PROPORTION X - x =-.080838 Z =-2.08932 P-VALUE = .0366 P-VALUE/2 = .0183 SD. ERROR = .0386912 State the proper conclusion if the test was conducted at α = .10.

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

Consider a test of $\mathrm { H } _ { 0 } : \mu = 10$ . For the following case, give the rejection region for the test in terms of the z-statistic: $\mathrm { H } _ { \mathrm { a } } : \mu > 10 , \alpha = 0.01$ A) $z > 2.33$ B) $z > 2.05$ C) $| z | > 2.575$ D) $| z | > 2.33$

The State Association of Retired Teachers has recently taken flak from some of its members regarding the poor choice of the association's name. The association's by-laws require that more than 60 percent of the association must approve a name change. Rather than convene a meeting, it is first desired to use a sample to determine if meeting is necessary. Suppose the association decided to conduct a test of hypothesis using the following null and alternative hypotheses: H0: p = 0.6 HA: p > 0.6 Define a Type I Error in the context of this problem.