Introductory Statistics Hypothesis Testing Review( Critical Value Approach)

Transcription

1 MULTIPLE CHOICE. Introductory Statistics Hypothesis Testing Review( Critical Value Approach) Determine the critical value(s) for a one-mean z-test. 1) A left-tailed test with α = A) ±1.96 B) ±2.054 C) D) ) 2) A revenue department is under orders to reduce the time small business owners spend filling out pension form ABC Previously the average time spent on the form was 6.3 hours. In order to test whether the time to fill out the form has been reduced, a sample of 53 small business owners who annually complete the form was randomly chosen, and their completion times recorded. The mean completion time for ABC-5500 form was 6.1 hours with a standard deviation of 2.6 hours. In order to test that the time to complete the form has been reduced, state the appropriate null and alternative hypotheses. A) H0: μ = 6.3 B) H0: μ = 6.3 C) H0: μ > 6.3 D) H0: μ = 6.3 Ha: μ > 6.3 Ha: μ < 6.3 Ha: μ < 6.3 Ha: μ 6.3 2) For the given hypothesis test, explain the meaning of a Type I error, a Type II error, or a correct decision as specified. 3) In the past, the mean running time for a certain type of flashlight battery has been 8.0 hours. The 3) manufacturer has introduced a change in the production method and wants to perform a hypothesis test to determine whether the mean running time has increased as a result. The hypotheses are: H : μ = 8.0 hours 0 : μ > 8.0 hours where μ is the mean running time of the new batteries. Explain the meaning of a Type I error. A) A Type I error would occur if, in fact, μ = 8.0 hours, but the results of the sampling lead to the conclusion that μ > 8.0 hours. B) A Type I error would occur if, in fact, μ > 8.0 hours, but the results of the sampling fail to lead to that conclusion. C) A Type I error would occur if, in fact, μ > 8.0 hours, but the results of the sampling lead to the conclusion that μ < 8.0 hours. D) A Type I error would occur if, in fact, μ = 8.0 hours, but the results of the sampling do not lead to rejection of that fact. 4) 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 34 minutes. The owner has randomly selected 19 customers and delivered pizzas to their homes. What hypotheses should the owner test to demonstrate that the pizza delivery will not be successful? A) H0: μ < 34 vs. Ha: μ = 34 B) H0: μ = 34 vs. Ha: μ < 34 C) H0: μ = 34 vs. Ha: μ > 34 D) H0: μ = 34 vs. Ha: μ 34 4) Find the rejection region for the specified hypothesis test. 5) Consider a test of H0: μ = 4. For the following case, give the rejection region for the test in terms of the z-statistic: Ha: μ < 4, α = 0.08 A) z < B) z > C) z < D) z < ) 1

2 Classify the conclusion of the hypothesis test as a Type I error, a Type II error, or a correct decision. 6) In the past, the mean running time for a certain type of flashlight battery has been 9.9 hours. The manufacturer has introduced a change in the production method and wants to perform a hypothesis test to determine whether the mean running time has increased as a result. The hypotheses are: H 0 : μ = 9.9 hours : μ > 9.9 hours where μ is the mean running time of the new batteries Suppose that the results of the sampling lead to nonrejection of the null hypothesis. Classify that conclusion as a Type I error, a Type II error, or a correct decision, if in fact the mean running time has increased. A) Correct decision B) Type I error C) Type II error 7) In the past, the mean running time for a certain type of flashlight battery has been 8.1 hours. The manufacturer has introduced a change in the production method and wants to perform a hypothesis test to determine whether the mean running time has increased as a result. The hypotheses are: H 0 : μ = 8.1 hours : μ > 8.1 hours where μ is the mean running time of the new batteries Suppose that the results of the sampling lead to rejection of the null hypothesis. Classify that conclusion as a Type I error, a Type II error, or a correct decision, if in fact the mean running time has not increased. A) Type II error B) Correct decision C) Type I error 6) 7) 8) 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: 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 = 18 B) z = ,000 C) z = 18 10,0002 D) z = 18 10,000 8) A sample mean, sample standard deviation, and sample size are given. Use the one-mean t-test to perform the required hypothesis test about the mean, μ, of the population from which the sample was drawn. Use the critical-value approach. 9) x = 39.4, s = 7.6, n = 15, H 0 : μ = 32.6, : μ 32.6, α = A) Test statistic: t = Critical values: t = ± Do not reject H0: μ = There is not sufficient evidence to support the claim that the mean is different from B) Test statistic: t = Critical values: t = ±1.96. Do not reject H0: μ = There is not sufficient evidence to support the claim that the mean is different from C) Test statistic: t = Critical values: t = ± Reject H0: μ = There is sufficient evidence to support the claim that the mean is different from D) Test statistic: t = Critical values: t = ±1.96. Reject H0: μ = There is sufficient evidence to support the claim that the mean is different from ) 2

3 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Preliminary data analyses indicate that it is reasonable to use a t-test to carry out the specified hypothesis test. Perform the t-test using the critical-value approach. 10) A large software company gives job applicants a test of programming ability and the 10) mean for that test has been 160 in the past. Twenty-five job applicants are randomly selected from a large university and they produce a mean score of 183 and standard deviation of 12. Use a 5% significance level to test whether the mean score for students from this university is greater than ) A light-bulb manufacturer advertises that the average life for its light bulbs is 900 hours. A random sample of 15 of its light bulbs resulted in the following lives in hours At the 10% significance level, do the data provide evidence that the mean life for the company's light bulbs differs from the advertised mean? 11) MULTIPLE CHOICE. The number of successes and the sample size are given for a simple random sample from a population. Decide whether using the one-proportion z-test is appropriate. 12) x = 25, n = 70, H0: p = 0.5, Ha: p ) A) Appropriate B) Not appropriate A hypothesis test is to be performed for a population proportion. For the given sample data and null hypothesis, compute the value of the test statistic, z = p^ - p0 p0(1 - p0)/n 13) A radio show producer believes that a new proposed format would be preferred by only 15% of their current listeners. In a random sample of 100 current listeners, 20% favored the new format. The null hypothesis is H0: p = A) B) C) D) ) Use the one-proportion z-test to perform the specified hypothesis test. Use the critical-value approach. 14) x = 22, n = 100, H0: p = 0.2, Ha: p 0.2, α = 0.01 A) z = 0.500; critical values = ±2.575; do not reject H0 B) z = 0.310; critical values = ±2.33; reject H0 C) z = 0.500; critical values = ±2.33; do not reject H0 D) z = 0.310; critical values = ±2.575; do not reject H0 14) 3

4 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 15) 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. 15) 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. 16) 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: x = hours, s = 110 hours. Conduct the test using α = ) MULTIPLE CHOICE. Use the one-proportion z-test to perform the specified hypothesis test. Use the critical-value approach. 17) x = 145, n = 165, H0: p = 0.92, Ha: p < 0.92, α = 0.10 A) z = -1.95; critical value = ; reject H0 B) z = -1.54; critical value = ; do not reject H0 C) z = -1.54; critical value = -1.28; do not reject H0 D) z = -1.95; critical value = -1.28; reject H0 17) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the one-proportion z-test to perform the required hypothesis test. Use the critical-value approach. 18) An airline's public relations department says that the airline rarely loses passengers' 18) luggage. It further claims that on those occasions when luggage is lost, 92% is recovered and delivered to its owner within 24 hours. A consumer group who surveyed a large number of air travelers found that only 145 out of 165 people who lost luggage on that airline were reunited with the missing items by the next day. At the 5% level of significance, do the data provide sufficient evidence to conclude that the proportion of times that luggage is returned within 24 hours is less than 0.92? 19) In 2000, the percentage of adults in a certain town who drove an SUV was 53%. In 2005, in a random sample of 100 people from this town, 45% said that they drive an SUV. At the 10% level of significance, do the data provide sufficient evidence to conclude that the percentage of adults in this town who drive an SUV has changed from the 2000 percentage of 53%? 19) 4

5 20) In a sample of 151 children selected randomly from one town, it is found that 31 of them suffer from asthma. At the 0.05 significance level, do the data provide sufficient evidence to conclude the proportion of all children in the town who suffer from asthma is different from 11%? 20) 5

6 Answer Key Testname: CH8 REVIEW CV APPROACH 1) D 2) B 3) A 4) C 5) C 6) C 7) C 8) D 9) C 10) H 0 : μ = 160. : μ > 160. α = 0.05 Test statistic: t = Critical value: t = Reject the null hypothesis. At the 5% significance level, there is sufficient evidence to conclude that the mean score for students from this university is greater than ) H 0 : μ = 900 hours : μ 900 hours α = 0.10 Test statistic: t = Critical values: t = ± Reject H 0 : μ = 900 hours. At the 10% significance level, there is sufficient evidence to conclude that the true mean life differs from the advertised mean of 900 hours. 12) A 13) D 14) A ) H0: μ = 70 vs. Ha: μ > 70; x = 79.98, s = 12.34; z = / Since 5.72 > 1.645, we reject the null hypothesis in favor of the alternative hypothesis. There is evidence to support the counselor's suspicions. 16) To determine if the mean life exceeds 800 hours, we test: H0: μ = 800 vs. Ha: μ > 800 The test statistic is z = x - μ 0 σ/ n x - μ = s/ n 110/ 121 = Since the test is greater than 1.645, H0 can be rejected. There is sufficient evidence to indicate the average life of the new bulbs exceeds 800 hours when testing at α = ) D 18) H0: p = 0.92; Ha: p < 0.92; α = 0.05 Test statistic: z = Critical value = Reject H0. At the 5% level of significance, the data provide sufficient evidence to conclude that the proportion of times that luggage is returned within 24 hours is less than ) H0: p = Ha: p α = 0.10 Test statistic: z = Critical values: z = ± Do not reject the null hypothesis. At the 10% level of significance, the data do not provide sufficient evidence to conclude that the percentage of adults in this town who drive an SUV has changed from the 2000 percentage of 53%. 6

7 Answer Key Testname: CH8 REVIEW CV APPROACH 20) H0: p = 0.11 Ha: p α = 0.05 Test statistic: z = Critical values: z = ±1.96. Reject H0. At the 5% level of significance, the data provide sufficient evidence to conclude that the proportion of all children in the town who suffer from asthma is different from 11%. 7