Chapters 7 and 8 Sample Exercises SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) A researcher collected data on the annual salary of doctors in Illinois.He wishes to construct a confidence interval for the mean annual salary. Which of the following are true? I. A 98% confidence interval will have a greater chance of including the true mean salary than a 95% confidence interval. II. A 98% confidence interval will be narrower than a 95% confidence interval. III. A 98% confidence interval requires the researcher to take a larger sample. 2) You have constructed a 95% confidence interval to estimate the mean number of people downloading music files from the Internet using a sample of 150 people. Which of the following statements are true? I. A 95% Confidence interval will be wider than a 99% confidence interval. II. Sampling a smaller number of people will increase the margin of error. III. Changing the sample size will change the value of t*. 3) A statistician in a mail order house wished to estimate how many mail order parcels prepared last Friday by the wrapping and packaging department were improperly packaged. He took a random sample of 3% of all the parcels prepared that day, had the sample parcels unwrapped and inspected, and found that 48 were improperly packaged. He reported that 1600 mail order parcels were improperly packaged on Friday. The statistician's report utilized 4) 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, $146,612). At what confidence level are the inferences derived from this information valid? Select the most appropriate answer. 5) A sample statistic tends to be an accurate estimate of its corresponding population parameter when 6) Both the sample mean and the sample proportion are ____________________ of the population mean and the population proportion, respectively. 7) In a survey of 500 residents, 300 were opposed to the use of the photo-cop for issuing traffic tickets. It is found that the point estimate for the mean is 60% with a standard error of 0.022. Construct the 95% confidence interval for the population proportion. 8) In monitoring lead in the air after an explosion at a battery factory, it is found that the amounts of lead (in ug/m3) in a 6 day period had a mean of 1.54 and a standard error of 1.91. Construct the 95% confidence interval for the population mean. Provide an appropriate response. 9) Once we've collected the data, how do we find a point estimate, representing our "best guess" for a parameter value? 10) Why is a point estimate alone not sufficiently informative? 11) What are the two properties of a good point estimator? 1 12) Explain what it means for a method of point estimation to be (a) unbiased, (b) efficient. 13) Explain how an interval estimate is more informative than a point estimate. 14) Is it true that the point estimate of a population mean must lie within the range of values defined by the corresponding confidence-interval estimate, regardless of the level of confidence achieved? Explain. ^ 15) You are planning to use a sample proportion p to estimate a population proportion, p. A sample size of 100 and a confidence level of 95% yielded a margin of error of 0.025. Which of the following will result in a larger margin of error? A: Increasing the sample size while keeping the same confidence level B: Decreasing the sample size while keeping the same confidence level C: Increasing the confidence level while keeping the same sample size D: Decreasing the confidence level while keeping the same sample size 16) 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? 17) Many people think that a national lobby's successful fight against gun control legislation is reflecting the will of a minority of Americans. A random sample of 4,000 citizens yielded 2290 who are in favor of gun control legislation. Find the point estimate for estimating the proportion of all Americans who are in favor of gun control legislation. Select the most appropriate answer. 18) In a survey of 500 residents, 300 were opposed to the use of the photo-cop for issuing traffic tickets. What is the best point estimate for the proportion of all residents opposed to the photo-cop use? 19) In monitoring lead in the air after an explosion at a battery factory, the following reading measured the amounts of lead (in ug/m3). What is the best point estimate for the population mean? Monday 5.40 Tuesday 1.10 Wednesday 0.42 Thursday 0.73 Friday 0.48 Saturday 1.10 Provide an appropriate response. 20) Suppose you have obtained a confidence interval for Вµ, but wish to obtain a greater degree of precision. Which of the following would result in a narrower confidence interval? I. Increasing the sample size while keeping the confidence level fixed II. Decreasing the sample size while keeping the confidence level fixed III. Increasing the confidence level while keeping the sample size fixed IV. Decreasing the confidence level while keeping the sample size fixed 21) A confidence interval for a population mean has a margin of error of 3.9. If the sample mean is 54.6, obtain the confidence interval. 2 22) Suppose that scores for men on an aptitude test have greater variability than scores for women on the same test. In other words, the population standard deviation is greater for the population of men than for the population of women. Based on a sample of size 50, a 95% confidence interval for the mean score, Вµ, of all women has a margin of error of 2.2. Which of the following confidence intervals will have a smaller margin of error? A. A 99% confidence interval for the mean score of women. Sample size = 50 B. A 95% confidence interval for the mean score of women. Sample size = 100 C. A 95% confidence interval for the mean score of men. Sample size = 50 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 23) In a poll of 278 voters in a certain city, 67% said that they backed a bill which would limit growth and development in their city. The margin of error in the poll was reported as 6 percentage points (with a 95% degree of confidence). Which statement is correct? A) There is not enough information to determine whether the margin of error is consistent with the sample size B) The reported margin of error is consistent with the sample size C) For the given sample size, the margin of error should be smaller than stated D) For the given sample size, the margin of error should be larger than stated E) The sample size is too small to achieve the stated margin of error 24) In a poll of 390 voters in a certain city, 77% said that they backed a bill which would limit growth and development in their city. The margin of error in the poll was reported as 5 percentage points (with a 95% degree of confidence). Which statement is correct? A) The reported margin of error is consistent with the sample size B) The stated margin of error could be achieved with a smaller sample size C) The sample size is too small to achieve the stated margin of error D) The sample size is too large to achieve the stated margin of error E) There is not enough information to determine whether the margin of error is consistent with the sample size SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Select the most appropriate answer. 25) A confidence interval is constructed by adding and subtracting a ____________________ to a given point estimate. 26) In practice a ____________________ is an estimated standard deviation of a sampling distribution. 27) In a survey of 500 residents, 300 were opposed to the use of the photo-cop for issuing traffic tickets. The standard error of the proportion is found to be 0.022. Find the margin of error that corresponds to a 95% confidence interval. 28) In monitoring lead in the air after an explosion at a battery factory, it is found that the amounts of lead (in ug/m3) in a 6 day period had a standard error of 1.91. Find the margin of error that corresponds to a 95% confidence interval. Provide an appropriate response. 29) Describe what it means if the margin of error for a 95% confidence interval for a population parameter equals 0.13. 3 30) In stating a confidence-interval estimate of a population mean, the level of confidence increases as the size of the interval . (increases, decreases) 31) In stating a confidence-interval estimate of a population mean, the level of confidence decreases as the size of the interval . (increases, decreases) 32) Suppose you have obtained a 95% confidence interval for Вµ. Which of the following statements is/are true regarding the relationship between precision and confidence level? Assume that the sample size is fixed. A. B. C. D. Increasing the confidence level to 99% will result in a narrower interval. Decreasing the confidence level to 90% will result in greater precision. Decreasing the precision will result in a higher confidence level. Increasing the precision will result in a higher confidence level. 33) In which of the following situations is it reasonable to use the z-interval procedure to obtain a confidence interval for the population mean? Assume that the population standard deviation is known. A. n = 10, the data contain no outliers, the variable under consideration is not normally distributed. B. n = 10, the variable under consideration is normally distributed. C. n = 18, the data contain no outliers, the variable under consideration is far from being normally distributed. D. n = 18, the data contain outliers, the variable under consideration is normally distributed. Find the standard error 34) In a sample of 200 observations, there were 80 positive outcomes. Find the standard error for the sample proportion. 35) Out of 400 trials, 60 turned out positive. Find the standard error for the sample proportion. 36) A poll of 163 voters resulted in 110 favorable responses. Find the standard error for the sample proportion. 37) In a survey of 550 T.V. viewers, 20% said they watch network news programs. Find the standard error for the sample proportion. 38) In a survey of 3200 T.V. viewers, 20% said they watch network news programs. Find the standard error for the sample proportion. Provide an appropriate response. 39) For estimating a population proportion, ^ ^ a. Find the standard error of p for n = 1000 when p = 0.10, 0.30, 0.50, 0.70, 0.90. b. Using these, explain why a confidence interval for a proportion close to 0.50 is wider than one close to 0 or 1 for the same sample size. Find the margin of error 40) In a survey of 5100 T.V. viewers, 40% said they watch network news programs. Find the margin of error for this survey if we want 95% confidence in our estimate of the percent of T.V. viewers who watch network news programs. 4 41) A survey found that 89% of a random sample of 1024 American adults approved of cloning endangered animals. Find the margin of error for this survey if we want 90% confidence in our estimate of the percent of American adults who approve of cloning endangered animals. 42) In a survey of 280 adults over 50, 75% said they were taking vitamin supplements. Find the margin of error for this survey if we want a 99% confidence in our estimate of the percent of adults over 50 who take vitamin supplements. 43) A recent poll of 1500 new home buyers found that 60% hired a moving company to help them move to their new home. Find the margin of error for this poll if we want 95% confidence in our estimate of the percent of new home buyers who hired movers. 44) A recent poll of 500 residents in a large town found that only 36% were in favor of a proposed referendum to build a new high school. Find the margin of error for this poll if we want 95% confidence in our estimate of the percent of residents in favor of this proposed referendum. 45) In a sample of 198 observations, there were 80 positive outcomes. Find the margin of error for the 95% confidence interval used to estimate the population proportion. 46) Out of 400 trials, 60 turned out positive. Find the margin of error for the 95% confidence interval used to estimate the population proportion. 47) In a survey of 2300 T.V. viewers, 690 said they watch network news programs. Find the margin of error for the 95% confidence interval used to estimate the population proportion. 48) A poll of 163 voters resulted in 110 favorable responses. Find the margin of error for the 95% confidence interval used to estimate the population proportion. 49) In a clinical test with 2171 subjects, 1214 showed improvement from the treatment. Find the margin of error for the 95% confidence interval used to estimate the population proportion. Provide an appropriate response. 50) A newspaper article about bilingualism in Canada states that its estimate for the proportion of all adult Canadians who are bilingual has a margin of error equal to 0.04. How could you explain what this means to someone who has not taken a statistics course? Use the given degree of confidence and sample data to construct a confidence interval for the population proportion. 51) A survey of 865 voters in one state reveals that 408 favor approval of an issue before the legislature. Construct a 95% confidence interval for the proportion of all voters in the state who favor approval. 52) Of 346 items tested, 12 are found to be defective. Construct a 98% confidence interval for the proportion of all such items that are defective. 53) When 293 college students are randomly selected and surveyed, it is found that 114 own a car. Construct a 99% confidence interval for the percentage of all college students who own a car. 54) Of 369 randomly selected medical students, 23 said that they planned to work in a rural community. Construct a 95% confidence interval for the percentage of all medical students who plan to work in a rural community. 5 Select the most appropriate answer. 55) When a higher confidence level is used, all other parts of the confidence interval held constant, 56) The width of a confidence interval estimate for a proportion will be Provide an appropriate response. 57) A city council votes to appropriate funds for a new civic auditorium. The mayor of the city threatens to veto this decision unless it can be shown that a majority of citizens would use it at least twice a year. The council commissions a poll of city residents. For a random sample of 400 residents, 230 say they would use the facility at least twice a year. Find a 95% confidence interval for the proportion of all residents of the town who would say they would use the proposed auditorium at least twice a year. Interpret the interval and advise the mayor. 58) You work for a credit card company. You are assigned to estimate the proportion of the accounts in which a customer applied for and received a card but never used it. For a random sample of 20 customers, 3 never used it. Find a 95% confidence interval for the population proportion. Can you conclude that fewer than half the people who received the credit card never used it? 59) A Gallup poll of 1013 people conducted in July 2002 for USA Today and CNN indicated that 41% of Americans said that they could trust most people. Can we conclude that less than half of all Americans feel this way? Explain your reasoning based on a 95% confidence level. 60) A poll in 2000 of 500 Canadians by the National Post asked whether marijuana should be legalized for medical purposes. 72% said definitely yes, 20% said probably, 2% said probably not, 5% said definitely not, and 2% had no opinion. a. Assuming that this was a random sample, construct a 95% confidence interval for the population proportion who would answer definitely yes or probably. Can you conclude that a majority of all Canadians would answer this way? Explain. b. Check that the sample size was large enough to construct the interval in (a). 61) In a poll of 553 voters in a certain city, 64% said that they backed a bill which would limit growth and development in their city. The margin of error in the poll was reported as 4 percentage points (with a 95% degree of confidence). Make a statement about the adequacy of the sample size for the given margin of error. 62) In a poll of 25,000 voters in a certain city, 74% said that they backed a bill which would limit growth and development in their city. The margin of error in the poll was reported as 2 percentage points (with a 95% degree of confidence). Make a statement about the adequacy of the sample size for the given margin of error. 63) After conducting a survey, a researcher wishes to cut the standard error (and thus the margin of error) to 1 of 3 its original value. How will the necessary sample size change? 64) In a survey of 1,000 television viewers, 40% said they watch network news programs. For a 90% confidence level, the margin of error for this estimate is 2.5%. If we want to be 92.5% confident, how will the margin of error change? 65) In a survey of 1,000 television viewers, 40% said they watch network news programs. For a 99% confidence level, the margin of error for this estimate is 3.99%. If we only want to be 90% confident, how will the margin of error change? 6 66) The real estate industry claims that it is the best and most effective system to market residential real estate. A survey of randomly selected home sellers in Illinois found that a 95% confidence interval for the proportion of homes that are sold by a real estate agent is 69% to 81%. Interpret the interval in this context Using the t-tables, software, or a calculator, report the t-score for the given confidence interval and degrees of freedom. 67) 90% confidence interval with df = 4. 68) 95% confidence interval with df = 15 69) 99% confidence interval with df = 24 70) A 95% confidence interval with df = 29 71) A 99% confidence interval from a sample of size 41 72) A 90% confidence interval from a sample of size 20. Provide an appropriate response. 73) Which of the following statements regarding t-curves is/are true? 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. 74) Suppose that you wish to obtain a confidence interval for a population mean. Under the conditions described below, should you use the z-interval, the t-interval, or neither? - The population standard deviation is unknown. - The population is normally distributed. - The sample size is small. 75) Suppose that you wish to obtain a confidence interval for a population mean. Under the conditions described below, should you use the z-interval, the t-interval, or neither? - The population standard deviation is unknown. - The data contain outliers. - The sample size is small. Find the requested value 76) A researcher wishes to estimate the mean resting heart rate for long-distance runners. A random sample of 12 long-distance runners yields the following heart rates, in beats per minute. 71 62 65 60 69 72 78 79 73 65 60 63 Use the data to obtain a point estimate of the mean resting heart rate for all long distance runners. 7 77) Physiologists often use the forced vital capacity as a way to assess a person's ability to move air in and out of their lungs. A researcher wishes to estimate the forced vital capacity of people suffering from asthma. A random sample of 15 asthmatics yields the following data on forced vital capacity, in liters. 5.1 5.3 4.4 3.9 4.3 3.3 3.6 4.3 3.4 3.1 3.2 3.5 4.8 4.0 5.1 Use the data to obtain a point estimate of the mean forced vital capacity for all asthmatics. 78) A researcher for a car insurance company wishes to estimate the mean annual premium that women aged 25-30 pay for their car insurance. A random sample of 16 women aged between 25 and 30 yields the following annual premiums, in dollars. 582 748 594 856 658 662 723 610 466 777 580 720 941 704 725 985 Use the data to obtain a point estimate of the mean annual premium for all women aged between 25 and 30. Round your answer to the nearest dollar. 79) A long-distance phone company wishes to estimate the mean duration of long-distance calls originating in Texas. A random sample of 15 long-distance calls originating in Texas yields the following call durations, in minutes. 2 3 1 4 3 8 34 27 13 29 1 19 12 2 37 Use the data to obtain a point estimate of the mean call duration for all long-distance calls originating in Texas. Determine the margin of error in estimating the population parameter. 80) Based on a sample of 39 randomly selected years, a 90% confidence interval for the mean annual precipitation in one city is from 42.8 inches to 45.2 inches. 81) Based on a sample of size 49, a 95% confidence interval for the mean score of all students on an aptitude test is from 64.3 to 69.7. 82) How tall is your average English classmate? To determine this, you measure the height of a random sample of 15 of your 100 fellow students, finding a 95% confidence interval for the mean height of 67.25 to 69.75 inches. 83) Alarmed at the rising gas prices in your town, you decide to estimate the average gas price for a gallon of regular gas. From your sample of 25 gas stations, you calculate a 90% confidence interval of ($1.99, $2.11) 84) How much fat do reduced fat cookies typically have? You take a random sample of 51 reduced-fat cookies and test them in a lab, constructing the following confidence interval: t-Interval for Вµ: with 90.00% Confidence, 2.3 < Вµ < 3.4 8 85) To determine the mean number of unpopped kernels in your favorite brand of microwave popcorn, you pop a random sample of 50 bags of popcorn and construct of 98% confidence interval of (12.5, 15.4). Construct the requested confidence interval from the supplied information. 86) Thirty randomly selected students took the statistics final. If the sample mean was 82 and the standard deviation was 12.2, construct a 99% confidence interval for the mean score of all students. 87) A sociologist develops a test to measure attitudes about public transportation, and 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4. Construct the 95% confidence interval for the mean score of all such subjects. 88) A savings and loan association needs information concerning the checking account balances of its local customers. A random sample of 14 accounts was checked and yielded a mean balance of $664.14 and a standard deviation of $297.29. Find a 98% confidence interval for the true mean checking account balance for local customers. 89) A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 246 milligrams with s = 11.7 milligrams. Construct a 95% confidence interval for the true mean cholesterol content of all such eggs. 90) A sample of 81 calculus students at a large college had a mean mathematics ACT score of 26 with a standard deviation of 6. Find a 95% confidence interval for the mean mathematics ACT score for all calculus students at this college. 91) Among a sample of 65 students selected at random from one high school, the mean number of siblings is 1.3 with a standard deviation of 1.1. Find a 95% confidence interval for the mean number of siblings for all students at this high school. 92) How much sugar do reduced sugar cookies typically have? You take a random sample of 51 reduced-sugar cookies and test them in a lab, finding a mean sugar content of 3.2 grams and a standard deviation of 1.1 grams of sugar. Create a 99% confidence interval for the mean grams of sugar. 93) How tall is your average English classmate? To determine this, you measure the height of a random sample of 15 of your 200 fellow students, finding a mean height of 68 inches and a standard deviation of 2.3 inches. Construct a 90% confidence interval for the mean height of your classmates. 94) A researcher wants to estimate the mean cholesterol level of people in his state.A random sample of 21 people yields a mean cholesterol level of 224 and a standard deviation of 12. Construct a 95% confidence interval. 95) A college math professor has office hours from 9:00 am to 10:30 am daily. A sample of waiting times to see the professor (in minutes) is 10, 12, 20, 15, 17, 10, 30, 28, 35, 28, 19, 27, 25, 22, 33, 37, 14, 21, 20, 23. Assuming = 7.84, find the 95% confidence interval for the population mean. Interpret the confidence interval. 96) Analysis of a random sample of 250 Virginia nurses produced a 95% confidence interval for the mean annual salary of $42,838 < Вµ < $49,691. 97) Data collected by child development scientists produced the following 90% confidence interval for the average age (in months) at which children say their first word: 10.4 < Вµ < 13.8. 9 98) A random sample of clients at a weight loss center were given a dietary supplement to see if it would promote weight loss. The center reported that the 100 clients lost an average of 46 pounds, and that a 95% confidence interval for the mean weight loss this supplement produced has a margin of error of В±9 pounds. 99) How many unpopped kernels are left when you pop a bag of microwave popcorn? Quality control personnel at ButterYum Popcorn take a random sample of 50 bags of popcorn. They pop each bag in a microwave and then count the number of unpopped kernels. The following interval is produced: t-interval for Вµ : with 99% Confidence, 11 < Вµ < 25 100) A researcher wants to estimate the mean cholesterol level of people in his city.A random sample of 21 people yields an average cholesterol level of 219, with a margin of error of В±12. Assume the researcher used a confidence level of 90%. Provide an appropriate response. 101) A 95% confidence interval for a population mean income goes from $30,000 to $32,000. Does this mean that it is impossible that the population mean is less than $30,000 or greater than $32,000? Explain. 102) For 2002 data from the GSS on subjects' reports of the number of sex partners they had in the last 12 months, a computer printout reports the following information for female respondents: Variable partners N Mean 1237 0.90 StDev 0.87 SE Mean 95.0% Cl 0.025 (0.85, 0.95) a. Based on the reported sample size and standard deviation, verify the reported value for the standard error. b. For the 1005 males, the mean was 1.30 with standard deviation 1.26. What two statistical factors cause a 95% confidence interval for females to be narrower than a 95% confidence interval for males? 103) The article "First Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students" (Journal of College Student Development (1999):599) reported that for students enrolled at a large research university the sample mean and standard deviation for high school grade point average were 3.73 and 0.45, respectively. Suppose the results were based on a random sample of 101 students at the university. Construct a 95% confidence interval for the mean high school GPA for students at this university. Interpret in context. 104) Recent findings have suggested that infant sex differences exist in behavioral and physiological reactions to stress. One study (M. Davis and E. Emory, Child Development, Vol. 66, 1995, pp. 14-27) evaluated changes in the heart rate for a sample of infants placed in a stressful situation. For the 15 female infants, a printout for the data on the change in heart rate shows: Variable change N df Mean StDev SE Mean 95.0% Cl 15 14 10.70 17.70 4.570 (0.90, 20.50) a. Show how the software obtained the value for "SE Mean." Explain what this represents. b. Explain how software obtained the value of df, and indicate which t-score was used in constructing the 95% confidence interval. c. From the confidence interval shown, can you conclude that the true mean change in heart rate is positive? Explain. d. Explain the implications of the term "robust" regarding the normality assumption made to conduct this analysis. 10 105) The variable EDUC in the 2002 General Social Survey asked "What is the highest grade that you finished and got credit for?" Of 2753 respondents, only 5 people said 0 years. You would like a 95% confidence interval for the population proportion with no formal education. a. b. Why is the ordinary large-sample confidence interval formula not valid? Construct a valid interval. Interpret in context. 106) A pollster wishes to estimate the true proportion of U.S. voters who oppose capital punishment. How many voters should be surveyed in order to be 95% confident that the true proportion is estimated to within 2%? 107) A survey of shoppers is planned to see what percentage use credit cards. Prior surveys suggest 63% of shoppers use credit cards. How many randomly selected shoppers must we survey in order to estimate the proportion of shoppers who use credit cards to within 4% with 95% confidence? 108) A university's administrator proposes to do an analysis of the proportion of graduates who have not found employment in their major field one year after graduation. In previous years, the percentage averaged 7%. He wants the margin of error to be within 4% at a 99% confidence level. What sample size will suffice? Use 2.575 as the critical value for a 99% confidence interval. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 109) The large sample confidence interval formula for estimating p should only be used when ^ ^ A) np 15 and n(1 - p) B) n 30 ^ C) np 15 ^ D) n(1 - p) ^ E) np 15 15 ^ 15 or n(1 - p) 15 110) What factor or factors affect the choice of the sample size when estimating Вµ or p? A) variability in the data B) confidence level C) desired precision D) financial cost E) All of the above. 111) When determining the sample size for estimating a population proportion for a given level of confidence and a desired margin of error, the closer to 0.50 that p is estimated to be A) the smaller the sample size required. B) has an undeterminable effect on the sample size required. C) the farther from 0.50 that 1 - p is estimated to be. D) has no effect on the sample size required. E) the larger the sample size required. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 112) A study will estimate the proportion of traffic deaths in Australia last year that were alcohol related. Determine the sample size for the estimate to be accurate to within 0.06 with probability 0.99. An earlier study estimated that the proportion was about 0.40. 11 113) An educator wants to estimate the proportion of school children in Boston who are living with only one parent. Since their report is to be published, they want a reasonably accurate estimate. However, since their funding is limited, they do not want to collect a larger sample than necessary. They hope to use a sample size such that, with probability 0.95, the error will not exceed 0.04. What sample size will ensure this, regardless of what sample proportion value occurs when they gather the sample? 114) The 1993 Australia Election Study asked 2380 subjects their opinion about the statement, "The smoking of marijuana should not be a criminal offence." 35% answered in the strongly agree or agree categories, 44% in the strongly disagree or disagree categories, and the rest were undecided. a. Conditional on not being undecided, find the percentage in the (i) strongly agree or agree categories, (ii) strongly disagree or disagree categories. b. Using (a) and the relevant sample size, determine whether you have sufficient evidence to conclude that, in the population, one of these two conditional percentages is higher than the other. Explain your reasoning, including assumptions. Find the sample size 115) A population is normal with a variance of 36. Suppose you wish to estimate the population mean Вµ. Find the sample size needed to assure with 68% confidence that the sample mean will not differ from the population mean by more than 4 units. 116) 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% confidence that the sample mean will not differ from the population mean by more than 2 units. 117) Weights of women in one age group are normally distributed with a standard deviation of 20 lb. A researcher wishes to estimate the mean weight of all women in this age group. Find how large a sample must be drawn in order to be 90% confident that the sample mean will not differ from the population mean by more than 3.5 lb. 118) Scores on a certain test are normally distributed with a variance of 20. A researcher wishes to estimate the mean score achieved by all adults on the test. Find the sample size needed to assure with 95% confidence that the sample mean will not differ from the population mean by more than 2 units. 119) The weekly earnings of students in one age group are normally distributed with a standard deviation of 81 dollars. A researcher wishes to estimate the mean weekly earnings of students in this age group. Find the sample size needed to assure with 98% confidence that the sample mean will not differ from the population mean by more than 5 dollars. 120) The drying times for a certain type of cement are normally distributed with a standard deviation of 73 minutes. A researcher wishes to estimate the mean drying time for this type of cement. Find the sample size needed to assure with 68% confidence that the sample mean will not differ from the population mean by more than 4 minutes. 121) The monthly credit card debts for individual accounts are normally distributed with a standard deviation of 61 dollars. A researcher wishes to estimate the mean monthly credit card debt for all individual accounts. Find the sample size needed to assure with 95% confidence that the sample mean will not differ from the population mean by more than 2 units. 12 122) 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. Provide an appropriate response. 123) A study is conducted of the distance that employees at a large company live from the company to find out whether people tend to have longer commutes than in the past. For a random sample of 36 employees, the mean distance is 5.3 miles and the standard deviation is 4.0. a. Find the margin of error for a 95% confidence interval for the mean distance from the factory of all employees. b. How large a sample would have been adequate if we merely needed a margin of error of 2.0? 124) To aid the establishment of personnel requirements, the director of a hospital needs to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. From a previous study, the standard deviation was found to be 5 admissions. If the director wishes to estimate the mean number of admissions per 24-hour period to within 1 admission with 99% confidence, what sample size should she choose? Express the null hypothesis. 125) Which could be the null hypothesis for the true proportion of fireflies unable to produce light? 126) Which is the null hypothesis for testing that the average (Вµ) mile per gallon of a new SUV called the Aquarius is better than 25. Examine the given statement, then identify whether the statement can be the null hypothesis, the alternative hypothesis or neither. 127) The mean income of workers who have majored in history is less than $25,000. 128) The percentage of viewers tuned to FOX News is equal to 85%. 129) The mean amount of Diet Dr. Pepper is at least 12 oz. 130) The more college courses a person has will lead to a higher salary. 131) The mean Вµ = 5.5. Determine the null and alternative hypotheses. 132) In the past, the mean running time for a certain type of radio battery has been 9.6 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 changed as a result. 133) A manufacturer claims that the mean amount of cola in its 16 ounce bottles is 16.1 ounces. A consumer advocacy group wants to perform a hypothesis test to determine whether the mean amount is actually less than this. 13 134) At one school, the average amount of time that tenth-graders spend watching television each week is 21.6 hours. The principal introduces a campaign to encourage the students to watch less television. One year later, the principal wants to perform a hypothesis test to determine whether the average amount of time spent watching television per week has decreased. 135) A health insurer has determined that the "reasonable and customary" fee for a certain medical procedure is $1200. They suspect that the average fee charged by one particular clinic for this procedure is higher than $1200. The insurer wants to perform a hypothesis test to determine whether their suspicion is correct. Provide an appropriate response. 136) Suppose the claim is in the alternate hypothesis. What form does your conclusion take? Suppose the claim is in the null hypothesis. What form does your conclusion take? MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 137) Which of the following would be an appropriate null hypothesis? A) The sample proportion is equal to 0.41. B) The population proportion is not equal to 0.41. C) The population proportion is equal to 0.41. D) The population proportion is less than 0.41. E) The sample proportion is less than 0.41. 138) Which of the following would be an appropriate null hypothesis? A) The population mean is equal to 3.4. B) The population mean is not equal to 3.4. C) The sample mean is greater than 3.4. D) The population mean is greater than 3.4. E) The sample mean is equal to 3.4. 139) Which of the following would be an appropriate alternative hypothesis? A) The sample proportion is less than 0.41. B) The sample proportion is not equal to 0.41. C) The population proportion is equal to 0.41. D) The sample proportion is equal to 0.41. E) The population proportion is less than 0.41. 140) Which of the following would be an appropriate alternative hypothesis? A) The sample mean is equal to 3.4. B) The population mean is equal to 3.4. C) The population mean is greater than 3.4. D) The sample mean is greater than 3.4. E) The sample mean is not equal to 3.4. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine the null and alternative hypotheses. 141) In 1990, the average duration of long-distance telephone calls originating in one town was 7.2 minutes. A long-distance telephone company wants to perform a hypothesis test to determine whether the average duration of long-distance phone calls has changed from the 1990 mean of 7.2 minutes. 14 142) A skeptical paranormal researcher claims that the proportion of Americans that have seen a UFO, p, is less than 5 in every one thousand. 143) The owner of a football team claims that the average attendance at games is over 81,000, and he is therefore justified in moving the team to a city with a larger stadium. 144) Hancock Motor Company claims that its new sedan, the Aquarius, will average better than 25 miles per gallon in the city. Use Вµ, the true average mileage of the Aquarius. 145) A researcher claims that 62% of voters favor gun control. 146) The percentage of viewers tuned to FOX News is equal to 85%. 147) The mean starting salary for students who have majored in statistics is $55,000. 148) The principal of a middle school claims that test scores of the seventh-graders at her school vary less than the test scores of seventh-graders at a neighboring school, which have variation described by = 14.7. 149) A cereal company claims that the mean weight of the cereal in its packets is at least 14 oz. Find the P-value for the indicated hypothesis test. 150) A medical school claims that more than 28% of its students plan to go into general practice. It is found that among a random sample of 130 of the school's students, 39% of them plan to go into general practice. Find the P-Value for a test of the school's claim. 151) In a sample of 88 children selected randomly from one town, it is found that 8 of them suffer from asthma. Find the P-value for a test of the claim that the proportion of all children in the town who suffer from asthma is equal to 11%. 152) In a sample of 47 adults selected randomly from one town, it is found that 9 of them have been exposed to a particular strain of the flu. Find the P-value for a test of the claim that the proportion of all adults in the town that have been exposed to this strain of the flu is 8%. 153) A manufacturer claims that fewer than 6% of its fax machines are defective. In a random sample of 100 such fax machines, 5% are defective. Find the P-value for a test of the manufacturer's claim. 154) An article in a journal reports that 34% of American fathers take no responsibility for child care. A researcher claims that the figure is higher for fathers in the town of Cheraw. A random sample of 225 fathers from Cheraw, yielded 97 who did not help with child care. Find the P-value for a test of the researcher's claim. 155) A nationwide study of American homeowners revealed that 65% have one or more lawn mowers. A lawn equipment manufacturer, located in Charlotte, feels the estimate is too low for households in Charlotte. Find the P-value for a test of the claim that the proportion with lawn mowers in Charlotte is higher than 65%. Among 497 randomly selected homes in Charlotte, 340 had one or more lawn mowers. 156) An airline claims that the no-show rate for passengers booked on its flights is less than 6%. Of 380 randomly selected reservations, 18 were no-shows. Find the P-value for a test of the airline's claim. 15 157) A random sample of 140 forty-year-old women contains 25% smokers. Find the P-value for a test of the claim that the percentage of forty-year-old women that smoke is 22%. 158) Find the P-value for a test of the claim that more than 50% of the people following a particular diet will experience increased energy. Of 100 randomly selected subjects who followed the diet, 47 noticed an increase in their energy level. Select the most appropriate answer. 159) Given Ha p0 . What is the P-value if the test statistics is calculated to be z = -0.12? 160) Given Ha p0 . What is the P-value if the test statistics is calculated to be z = 7.91? 161) Given Ha p0 . What is the P-value if the test statistics is calculated to be z = 0.58? Explain what the P-value means in the given context. 162) A state university wants to increase its retention rate of 4% for graduating students from the previous year. After implementing several new programs during the last two years, the university reevaluates its retention rate and comes up with a P-value of 0.075. What is reasonable to conclude about the new programs using = 0.05? 163) The federal guideline for smog is 12% pollutants per 10,000 volume of air. A metropolitan city is trying to bring its smog level into federal guidelines. The city comes up with a new policy where city employees are to use city transportation to and from work. A local environmental group does not think the city is doing enough and no real decrease will occur. An independent agency, hired by the city, runs its tests and comes up with a P-value of 0.055. What is reasonable to conclude about the new strategy using = 0.05? 164) A weight loss center provided a loss for 72% of its participants. The center's leader decides to test a new weight loss strategy to see if it's better and receives a P-value of 0.23. What is reasonable to conclude about the new strategy using = 0.1? For the given sample data and null hypothesis, compute the value of the test statistic, z 165) A claim is made that the proportion of children who play sports is less than 0.5, and the sample statistics include 1933 subjects with 30% saying that they play a sport. 166) The claim is that the proportion of drowning deaths of children attributable to beaches is more than 0.25, and the sample statistics include n = 615 drowning deaths of children with 30% of them attributable to beaches. 167) In a school district with 10,000 high school students, 1200 students completed a special class designed to improve their math skills. 708 of these scored better than the district-wide median on a standardized math exam. Does the special class have some value? The hypotheses are H : p = 0.5, H : p > 0.5, where p is the 0 a proportion of all those taking the special class who score better than the district-wide median. 168) A research group wants to determine whether the proportion of car accidents caused by drivers using cell phones has changed from the previous value of 13%. They obtained 10,000 auto accident reports and found that 14% were caused by drivers using cell phones. The hypotheses are H : p = 0.13, H : p 0.13, where p is the 0 a proportion of car accidents caused by drivers using cell phones. 169) Out of 199 observations, 50% were successes. H0: p = 0.43. 16 170) 410 people were asked if they were satisfied with their jobs. 37% said they were. H0: p = 0.30 171) A drug company claims that over 80% of all physicians recommend their drug. 1200 physicians were asked if they recommend the drug to their patients. 30% said yes. H0: p = 0.8 172) A radio show producer believes that a new proposed format would be preferred by only 20% of their current listeners. A survey of 100 current listeners showed that 17% favored the new format. Does the producer know his business? H0 : p = 0.20 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 173) A researcher claims that less than 20% of American adults are allergic to pollen. In a random sample of 100 adults, 15% indicate that they have such an allergy. Calculate the test statistic z for the population proportion. A) -1.25 B) 1.96 C) 1.645 D) 1.25 E) None of the above. 174) A research company claims that more than 55% of American regularly watch FOX News Watch. You decide to test this claim and ask a random sample of 425 Americans if they watch this program regularly. Of the 425, 255 respond yes. Calculate the test statistic z for the population proportion. A) 1.645 B) 2.07 C) -1.96 D) -2.07 E) None of the above. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. State conclusion to significance test in terms of the null hypothesis 175) According to a recent poll the percentage of Americans who would vote for the incumbent president is 53%. If a random sample of 100 people in New York results in 45% who would vote for the incumbent, test the claim that the percentage of people in New York who would vote for the incumbent president is different from 53%. Use a 0.10 significance level. H0 : p = 0.53. Ha : p 0.53. = 0.10 Test statistic: z = -1.60. P-Value = 0.1090 State your conclusion about H0 . 17 176) An article in a journal reports that 34% of American fathers take no responsibility for child care. A researcher claims that the figure is higher for fathers in the town of Cheraw. A random sample of 233 fathers from Cheraw yielded 96 who did not help with child care. Do the data provide sufficient evidence to conclude that in Littleton the proportion is higher than 0.34? Use a 0.05 significance level. H0 : p = 0.34 Ha : p > 0.34. = 0.05 Test statistic: z = 2.32. P-Value = 0.0102 State your conclusion in terms of the H0. 177) In a sample of 88 adults selected randomly from one town, it is found that 6 of them have been exposed to a particular strain of the flu. At the 0.01 significance level, test the claim that the proportion of all adults in the town that have been exposed to this strain of the flue differs from the nationwide percentage of 8%. H0 : p = 0.08 Ha : p 0.08. = 0.01 Test statistic: z = -0.41. P-Value = 0.6828 State your conclusion in terms of H0 . 178) A nationwide study of American homeowners revealed that 65% have one or more lawn mowers. A lawn equipment manufacturer, located in Omaha, feels the estimate is too low for households in Omaha. Can the value 0.65 be rejected if a survey of 490 homes in Omaha yields 336 with one or more lawn mowers? Use = 0.05. H0 : p = 0.65. Ha : p > 0.65. Test statistic: z = 1.66. P-value: p = 0.0485. State your conclusion in terms of H0 . Provide an appropriate response. 179) The county health department has concerns about the chlorine level of 0.4% mg/mL at a local water park increasing to an unsafe level. The water department tests the hypothesis that the local water park's chlorine proportions have remained the same, and find a P-value of 0.005. Provide an appropriate conclusion Perform a significance test for a population proportion using the P-value approach. 180) A manufacturer considers his production process to be out of control when defects exceed 3%. In a random sample of 80 items, the defect rate is 5% but the manager claims that this is only a sample fluctuation and production is not really out of control. At the 0.01 level of significance, do the data provide sufficient evidence that the percentage of defects exceeds 3%? Perform a significance test for a population proportion using the critical value approach. 181) A supplier of 3.5" disks claims that no more than 1% of the disks are defective. In a random sample of 600 disks, it is found that 3% are defective, but the supplier claims that this is only a sample fluctuation. At the 0.01 level of significance, do the data provide sufficient evidence that the percentage of defects exceeds 1%? 182) A poll of 1000 adult Americans reveals that 48% of the voters surveyed prefer the Democratic candidate for the presidency. At the 0.05 level of significance, do the data provide sufficient evidence that the percentage of voters who prefer the Democrat is less than 50%? Provide an appropriate response. 183) What are the assumptions for a significance test about a population proportion? 18 184) About 4% of Americans are vegetarians. For a random sample of 400 students at Texas A&M University, suppose that none are vegetarians. a. Set up null and alternative hypotheses for testing whether the proportion of vegetarians is the same or different at that university than nationwide. Check that the sample size is large enough for the test. b. Find the sample proportion, standard error, and test statistic. c. Find the P-value. Is there strong evidence that the proportion of vegetarians at the university differs from 0.04? 185) Does a majority or a minority of American workers think that they have a healthy balance between work and personal life? A poll of 1626 workers by an online job board True Careers reported that 70% don't think there's a healthy balance. Assuming this was a random sample, use a significance test to answer this question. Interpret, and explain how you can make a decision using a 0.10 significance level. 186) A multiple-choice test question has four possible responses. It first occurs on an exam taken by 400 students. The designers want to test whether more people answer the question correctly than would be expected due to chance (that is, if everyone randomly guessed the correct answer). a. b. c. Define notation and set up hypotheses to determine this. 125 of 400 students got the correct response. Find the P-value, and interpret. Make a decision about H0 , using a significance level of 0.05. Based on this decision, what can you conclude about the parameter? 187) A Pew Research Center poll (May 14, 2003) asked the question, "In order to overcome past discrimination, do you favor or oppose affirmative action programs designed to help blacks, women and other minorities get better jobs and education?" Of a random sample of 1201 adults in the U.S., 63% said favor, 29% said oppose, and 8% said don't know. Let p denote the population proportion who favored affirmative action. Conduct all five steps of a test of H0 : p = 0.50 against Ha : p 0.50. Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic t. 188) Test the claim that for the population of female college students, the mean weight is given by Вµ = 132 lb. Sample data are summarized as n = 20, x = 137 lb, and s = 14.2 lb. Use a significance level of = 0.1. Find the test statistic t. 189) Test the claim that for the adult population of one town, the mean annual salary is given by Вµ = $30,000. Sample data are summarized as n = 17, x = $22,298, and s = $14,200. Use a significance level of = 0.05. Find the test statistic t. 190) Test the claim that the mean lifetime of car engines of a particular type is greater than 220,000 miles. Sample data are summarized as n = 23, x = 226,450 miles, and s = 11,500 miles. Use a significance level of = 0.01. Find the test statistic t. 191) Test the claim that the mean age of the prison population in one city is less than 26 years. Sample data are summarized as n = 25, x = 24.4 years, and s = 9.2 years. Use a significance level of = 0.05. Find the test statistic t. 192) Test the claim that for the population of history exams, the mean score is 80. Sample data are summarized as n = 16, x = 84.5, and s = 11.2. Use a significance level of = 0.01. Find the test statistic t. 19 Provide an appropriate response. 193) In an advertising claim, an online computer support company claim that their mean call-back time is less than 30 minutes. A random sample of 36 calls has a sample mean of 28.5 minutes and a standard deviation of 3.5 minutes. Calculate the test statistic t for this for the population mean. 194) A soft drink company claims the mean caffeine content of its HoP ToP soda is 40 milligrams per one 8-ounce bottle. To verify this claim, a random sample of 30 bottles is found to have a mean caffeine content of 39.2 milligrams with a standard deviation of 7.5 milligrams. Calculate the test statistic t for this for the population mean. Assume that a simple random sample has been selected from a normally distributed population. State the final conclusion. 195) Test the claim that for the population of female college students, the mean weight is given by Вµ = 132 lb. Sample data are summarized as n = 20, x = 137 lb, and s = 14.2 lb. Use a significance level of H0 : Вµ = 132 Ha : Вµ 132 = 0.1. State your conclusion about H0 . 196) Test the claim that for the adult population of one town, the mean annual salary is given by Вµ = $30,000. Sample data are summarized as n = 17, x = $22,298, and s = $14,200. Use a significance level of = 0.05. H0 : Вµ = 30,000 H0 : Вµ 30,000 State your conclusion about H0 . 197) Test the claim that the mean lifetime of car engines of a particular type is greater than 220,000 miles. Sample data are summarized as n = 23, x = 226,450 miles, and s = 11,500 miles. Use a significance level of = 0.01. H0 : Вµ = 220,000 Ha: Вµ > 220,000 State your conclusion about H0 . 198) Test the claim that the mean age of the prison population in one city is less than 26 years. Sample data are summarized as n = 25, x = 24.4 years, and s = 9.2 years. Use a significance level of = 0.05. H0 : Вµ = 26 Ha : Вµ < 26 State your conclusion about H0 . 199) Test the claim that for the population of history exams, the mean score is 80. Sample data are summarized as n = 16, x = 84.5, and s = 11.2. Use a significance level of = 0.01. H0 : Вµ = 80 Ha : Вµ 80 State your conclusion about H0 . MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 200) The test statistic for testing H0 : Вµ = 100 against Ha : Вµ 100 was t = 3.3, with P-value 0.001. Then, A) there is not strong evidence that Вµ = 100. B) this must be wrong, because a large t test statistic must have a large P-value. C) there is strong evidence that Вµ > 100. D) there is not strong evidence that Вµ < 100. E) there is not enough information here to draw a conclusion. 20 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 201) What are the assumptions for a significance test about a population mean? 202) Recent findings have suggested that neonatal sex differences exist in behavioral and physiological reactions to stress. One study (M. Davis and E. Emory, Child Development, Vol. 66, 1995, pp. 14-27) evaluated changes in the heart rate for a sample of infants placed in a stressful situation. For the 15 female infants, the following is a printout for the data on the change in heart rate. Variable CHANGE a. b. c. d. Number of Cases Mean SD SE of Mean t-value df 2-Tail Sig 15 10.70 17.70 4.570 2.341 14 0.0346 State the hypotheses. State the test statistic. State the P-value. Interpret the P-value in context. 203) In response to the statement, "A preschool child is likely to suffer if his or her mother works," the response categories (Strongly agree, Agree, Disagree, Strongly disagree) had counts (155, 611, 863, 180) for the 1809 responses in a recent General Social Survey. With scores (2, 1, -1, -2) for the four categories, software reported the following results: Test of mu = 0 vs mu not = 0 Variable Opinion N 1809 Mean -0.1669 StDev 1.236 Variable Opinion 95.0% CI T (-0.224, -0.110) -5.735 SE Mean 0.0291 P 0.000 a. Define Вµ and set up null and alternative hypotheses to test whether the population mean differs from the neutral value, 0. b. Carry out the significance test, discussing assumptions and showing how software got the results stated in the table for the standard error and the test statistic value. 204) A sample was used to obtain a 95% confidence interval for the population mean age of graduate students at a large university. The 95% confidence interval for Вµ was (23, 27). If the same sample had been used to test the null hypothesis that the population mean age is equal to 28 versus the alternative hypothesis that the population mean age differs from 28, would the null hypothesis be rejected at = 0.05? Explain. Classify the significance test as two-tailed, left-tailed, or right-tailed. 205) A manufacturer claims that the mean amount of juice in its 16 ounce bottles is 16.1 ounces. A consumer advocacy group wants to perform a significance test to determine whether the mean amount is actually less than this. 206) In the past, the mean running time for a certain type of flashlight battery has been 9.3 hours. The manufacturer has introduced a change in the production method and wants to perform a significance test to determine whether the mean running time has changed as a result. 21 207) At one school, the average amount of time that ninth-graders spend watching television each week is 21.6 hours. The principal introduces a campaign to encourage the students to watch less television. One year later, the principal wants to perform a significance test to determine whether the average amount of time spent watching television per week has decreased from the previous mean of 21.6 hours. 208) A health insurer has determined that the "reasonable and customary" fee for a certain medical procedure is $1500. They suspect that the average fee charged by one particular clinic for this procedure is higher than $1500. The insurer wants to perform a significance test to determine whether their suspicion is correct. 209) In 1990, the average duration of long-distance telephone calls originating in one town was 15.3 minutes. A long-distance telephone company wants to perform a significance test to determine whether the average duration of long-distance phone calls has changed from the 1990 mean of 15.3 minutes. 210) The owner of a football team claims that the average attendance at games is over 80,000, and he is therefore justified in moving the team to a city with a larger stadium. An independent investigator will conduct a significance test to determine whether his claim is accurate. 211) The manufacturer of a refrigerator system for beer kegs produces refrigerators that are supposed to maintain a true mean temperature, Вµ, of 46В°F, ideal for a certain type of German pilsner. The owner of the brewery does not agree with the refrigerator manufacturer, and will conduct a significance test to determine whether the true mean temperature differs from this value. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 212) If an agronomist wishes to determine whether there is evidence that the average number of bales of cotton produced in a certain county equals 500, A) a two-tailed test should be used. B) a left-tailed test should be used. C) either a one-sided or a two-sided test could be used with equivalent results. D) a right-tailed test should be used. E) more information is necessary to determine what type of test should be used. 213) If an agronomist wishes to determine whether there is evidence that the average number of bales of cotton produced in a certain county exceeds 500, A) a two-tailed test should be used. B) a right-tailed test should be used. C) either a one-sided or a two-sided test could be used with equivalent results. D) a left-tailed test should be used. E) more information is necessary to determine what type of test should be used. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. For the given significance test, explain the meaning of a Type I error, a Type II error, or a correct decision as specified. 214) In the past, the mean running time for a certain type of radio battery has been 9.8 hours. The manufacturer has introduced a change in the production method and wants to perform a significance test to determine whether the mean running time has increased as a result. The hypotheses are: H : Вµ = 9.8 hours 0 H : Вµ > 9.8 hours a Explain the meaning of a Type I error. 22 215) A manufacturer claims that the mean amount of juice in its 16 ounce bottles is 16.1 ounces. A consumer advocacy group wants to perform a significance test to determine whether the mean amount is actually less than this. The hypotheses are: H : Вµ = 16.1 ounces 0 H : Вµ < 16.1 ounces a Explain the meaning of a Type I error. 216) At one school, the average amount of time that tenth-graders spend watching television each week is 21.6 hours. The principal introduces a campaign to encourage the students to watch less television. One year later, the principal wants to perform a significance test to determine whether the average amount of time spent watching television per week has decreased. The hypotheses are: H : Вµ = 21.6 hours 0 H : Вµ < 21.6 hours a Explain the meaning of a correct decision. 217) A health insurer has determined that the "reasonable and customary" fee for a certain medical procedure is $1200. They suspect that the average fee charged by one particular clinic for this procedure is higher than $1200. The insurer wants to perform a significance test to determine whether their suspicion is correct. The hypotheses are: H : Вµ = $1200 0 H : Вµ > $1200 a Explain the meaning of a correct decision. Provide an appropriate response. 218) A state university wants to increase its retention rate of 4% for graduating students from the previous year. After implementing several new programs during the last two years, the university reevaluated its retention rate. Identify the Type I error in this context. 219) The federal guideline for smog is 12% pollutants per 10,000 volume of air. A metropolitan city is trying to bring its smog level into federal guidelines. The city comes up with a new policy where city employees are to use city transportation to and from work. A local environmental group does not think the city is doing enough and no real decrease will occur. Identify the Type I error in this context. 220) A weight loss center provided a loss for 72% of its participants. The center's leader decides to test a new weight loss strategy. Identify the Type I error in this context. 221) The U.S. Department of Labor and Statistics released the current unemployment rate of 5.3% for the month in the U.S. and claims the unemployment has not changed in the last two months. However, the state's statistics reveal that there is a reduction in the U.S. unemployment rate. Identify the Type II error in this context. 23 222) A social psychologist plans to conduct an experiment with a random sample of 49 children from a school district. Before conducting the experiment, the psychologist checks how this sample compares to national norms on several variables. The IQ scores for the 49 children have x = 103 and s = 14. Nationally, the population mean IQ equals 100. Is it plausible that the mean Вµ of the population of children in the school district from which these students were sampled equals 100? a. Show all five steps of a test of H0 : Вµ = 100 against Ha : Вµ 100 using a significance level of 0.05. b. If the decision in (a) is an error, what type of error is it, Type I or Type II? Why? c. What conclusion applies for each of the following significance levels: (i) = 0.20, (ii) = 0.10, (iii) Why is = 0.20 rare in practice? = 0.01. 223) The mean score for all U.S. high school seniors taking the SAT college entrance exam equals 500. A study is conducted to see whether a different mean applies to Canadian seniors. For a random sample of 100 Canadian seniors, suppose the mean and standard deviation on this exam equal 508 and 100. a. b. Set up hypotheses for a significance test, and compute the test statistic. The P-value is 0.43. Interpret it, and make a decision about H0, using a significance level of 0.05. c. If the decision in (b) was in error, what type of error is it? d. A 95% confidence interval for Вµ is (488.2, 527.8). Show the correspondence between the decision in the test and whether 500 falls in this confidence interval. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 224) Failing to reject a false H0 A) is a Type I error. B) is a correct decision. C) has probability 1 - of occurring. D) has probability of occurring. E) is a Type II error. 225) Rejecting a true H0 A) is a correct decision. B) has probability 1 - of occurring. C) is a Type II error. D) is a Type I error. E) has probability of occurring. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Classify the conclusion of the significance test as a Type I error, a Type II error, or No error. 226) A manufacturer claims that the mean amount of juice in its 16 ounce bottles is 16.1 ounces. A consumer advocacy group wants to perform a significance test to determine whether the mean amount is actually less than this. The hypotheses are: H : Вµ = 16.1 ounces 0 H : Вµ < 16.1 ounces a Suppose that the results of the sample 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 amount of juice, Вµ, is less than 16.1 ounces. 24 227) In the past, the mean running time for a certain type of flashlight battery has been 9.6 hours. The manufacturer has introduced a change in the production method and wants to perform a significance test to determine whether the mean running time has increased as a result. The hypotheses are: H : Вµ = 9.6 hours 0 H : Вµ > 9.6 hours a Suppose that the results of the sample 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. 228) In the past, the mean running time for a certain type of flashlight battery has been 9.5 hours. The manufacturer has introduced a change in the production method and wants to perform a significance test to determine whether the mean running time has increased as a result. The hypotheses are: H : Вµ = 9.5 hours 0 H : Вµ > 9.5 hours a Suppose that the results of the sample 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. 229) At one school, the average amount of time that tenth-graders spend watching television each week is 21 hours. The principal introduces a campaign to encourage the students to watch less television. One year later, the principal wants to perform a significance test to determine whether the average amount of time spent watching television per week has decreased. The hypotheses are: H : Вµ = 21 hours 0 H : Вµ < 21 hours a Suppose that the results of the sample 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 amount of time, Вµ, spent watching television has not decreased. 230) A health insurer has determined that the "reasonable and customary" fee for a certain medical procedure is $1200. They suspect that the average fee charged by one particular clinic for this procedure is higher than $1200. The insurer wants to perform a significance test to determine whether their suspicion is correct. The hypotheses are: H : Вµ = $1200 0 H : Вµ > $1200 a Suppose that the results of the sample 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 average fee charged by the clinic is $1200 . 231) In 1990, the average duration of long-distance telephone calls originating in one town was 9.4 minutes. A long-distance telephone company wants to perform a significance test to determine whether the average duration of long-distance phone calls has changed from the 1990 mean of 9.4 minutes. The hypotheses are: H : Вµ = 9.4 minutes 0 H : Вµ 9.4 minutes a Suppose that the results of the sample 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 duration of long-distance phone calls has changed from the 1990 mean of 9.4 minutes. 25 232) A man is on trial accused of murder in the first degree. The prosecutor presents evidence that he hopes will convince the jury to reject the hypothesis that the man is innocent. This situation can be modeled as a significance test with the following hypotheses: H : The defendant is innocent. 0 H : The defendant is guilty. a Suppose that the null hypothesis is rejected; i.e., the defendant is found guilty. Classify that conclusion as a Type I error, a Type II error, or a correct decision, if in fact the defendant is innocent. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Select the most appropriate answer. 233) For a given level of significance, increasing the sample size will ____________________ the probability of committing a Type I error. A) not affect B) always decrease C) always increase D) sometimes increase E) sometimes decrease 234) The probability of Type II error increases when A) the true parameter value moves farther away from the parameter value given in H0 . B) the true parameter value moves closer to the parameter value given in H0 . C) the probability of Type I error decreases. D) both a. and b. E) both a. and c. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 235) When would = 0.01 be preferred to = 0.05? 26