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Chapter 7 - Practice Problems 3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the margin of error in estimating the population mean, μ. 1) Based on a sample of 36 randomly selected years, a 90% confidence interval for the mean annual precipitation in one city is from 46.3 inches to 49.7 inches. Find the margin of error. 1) A) There is not enough information to find the margin of error. B) 3.4 inches C) 1.7 inches D) 0.47 Find the necessary sample size. 2) Scores on a certain test are normally distributed with a variance of 14. A researcher wishes to estimate the mean score achieved by all adults on the test. Find the sample size needed to assure with 98 percent confidence that the sample mean will not differ from the population mean by more than 2 units. A) 11 B) 39 C) 20 D) 267 3) You wish to estimate the mean weight of machine components of a certain type and you require a 92% degree of confidence that the sample mean will be in error by no more than 0.008 g. Find the sample size required. A pilot study showed that the population standard deviation is estimated to be 0.06 g. A) 11 B) 173 C) 14 D) 112 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 4) Based on a sample of size 25, a researcher obtains an estimate of 64.4 inches for the mean height of all women aged 30-40. At the 95% confidence level, the margin of error is 1.2 inches. Do you agree with the interpretation below? If not, explain why not and give a correct interpretation. 4) Researcherʹs interpretation: ʺWe can be 95% confident that the height of a randomly selected woman will differ from 64.4 inches by at most 1.2 inches.ʺ 5) A sample mean is used to estimate a population mean. To obtain a margin of error of 1.5 at a confidence level of 95%, a sample size of 120 is needed. Would the required sample size be larger or smaller if the researcher wished to 5) (a) increase the confidence level while keeping the same margin of error? (b) decrease the margin of error while keeping the same confidence level? Explain your answers. 6) Compare the basic properties of t-curves with the basic properties of the standard normal curve. In what ways are t-curves similar to the standard normal curve? In what ways are they different? 1 2) 6) 3) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) Which of the following statements regarding t-curves is/are true? 7) A. The total area under a t-curve with 10 degrees of freedom is greater than the area under the standard normal curve. B. The t-curve with 10 degrees of freedom is flatter and wider than the standard normal curve. C. The t-curve with 10 degrees of freedom more closely resembles the standard normal curve than the t-curve with 20 degrees of freedom. A) A B) C C) B D) B, C Solve the problem. 8) A 90% confidence interval for the mean percentage of airline reservations being canceled on the day of the flight is (1.3%, 5.1%). What is the point estimate of the mean percentage of reservations that are canceled on the day of the flight? A) 2.55% B) 1.90% C) 3.8% D) 3.20% 9) Forty-five CEOs from the electronics industry were randomly sampled and a 95% confidence interval for the average salary of all electronics CEOs was constructed. The interval was ($129,997, $149,612). At what confidence level are the inferences derived from this information valid? A) 5% B) 47.5% C) 0.95% 8) 9) D) 95% 10) Forty-five CEOs from the electronics industry were randomly sampled and a 90% confidence interval for the average salary of all electronics CEOs was constructed. The interval was ($95,136, $110,372). Give a practical interpretation of the interval above. 10) A) 90% of the electronics industry CEOs have salaries that fall between $95,136 to $110,372. B) We are 90% confident that the mean salary of all the electronics industry CEOs falls in the interval $95,136 to $110,372. C) 90% of the sampled CEOs salaries fell in the interval $95,136 to $110,372. D) We are 90% confident that the mean salary of the sampled CEOs falls in the interval $95,136 to $110,372. 11) Forty-five CEOs from the electronics industry were randomly sampled and a 99% confidence interval for the average salary of all electronics CEOs was constructed. The interval was ($142,281, $153,625). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. What will result in a reduced interval width? A) Decrease the sample size and increase the confidence level. B) Decrease the sample size and decrease the confidence level. C) Increase the sample size and decrease the confidence level. D) Increase the sample size and increase the confidence level. 2 11) 12) At Great State University there are many sections of Introductory Statistics. All students are given a list of selected homework problems for each section. Some students choose to do the homework and some do not. The scores of the first test from a random sample of 100 students who do their homework were analyzed. The scores had a mean of 82.4 with a standard deviation of 5.6. The true mean score of the students doing their homework is 72.1. Is there evidence at the 95% confidence level that the true mean test score for the students doing their homework differs from 72.1? 12) A) Yes. 72.1 is more than one standard deviation from 82.4. B) No. 72.1 is within the 95% confidence interval. C) Yes. 72.1 falls outside of the 95% confidence interval. D) No. 72.1 is outside of the 95% confidence interval, but we need to also find the 95% confidence level of the scores of the students neglecting their homework. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 13) The numbers of advertisements seen or heard in one week for 30 randomly selected people in the United States are listed below. Construct a 95% confidence interval for the true mean number of advertisements. 598 481 734 494 298 588 441 135 590 595 846 540 728 764 673 690 317 727 684 649 545 486 732 486 735 582 702 13) 808 677 703 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 14) Find the value t0 such that the following statement is true: P(-t0 ≤ t ≤ t0 ) = .99 where df = 9. A) 2.2821 B) 3.250 C) 1.833 14) D) 2.262 15) Suppose a 98% confidence interval for μ turns out to be (1,000, 2,100). If this interval was based on a sample of size n = 23, explain what assumptions are necessary for this interval to be valid. A) The population must have an approximately normal distribution. B) The sampling distribution of the sample mean must have a normal distribution. C) The sampling distribution must be biased with 22 degrees of freedom. D) The population must have an approximate t distribution. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 16) A marketing research company needs to estimate the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 98% confidence interval was calculated to be ($2,181,260, $5,836,180). Based on the interval above, do you believe the average total compensation of CEOs in the service industry is $1,500,000? 3 16) 15) Answer Key Testname: CH 7 SET 3 1) 2) 3) 4) C C B The researcherʹs interpretation is not correct since the margin of error refers to the difference between the estimate and the true mean, not to the difference between the estimate and the height of an individual woman. The correct interpretation is: We can be 95% confident that the estimate of 64.4 inches differs from the true mean by at most 1.2 inches. 5) In both cases, a larger sample would be needed. 6) For both t-curves and the standard normal curve: - The total area under the curve is 1. - The curve is bell-shaped. - The curve is symmetrical about 0. - The curve extends indefinitely in both directions, approaching, but never touching, the horizontal axis as it does so. The t-curves are flatter and wider than the standard normal curve, though as the number of degrees of freedom increases, the t-curves looks increasingly like the standard normal curve. 7) C 8) D 9) D 10) B 11) C 12) C 13) (543.8, 658. 0) 14) B 15) A 16) No, the average total compensation is not $1,500,000 and we are 98% sure of this statement. This is because the value $1,500,000 is not contained in the 98% confidence interval for μ. 4