Problem 1A consumer testing agency plans to calculate a \(99 \%\) confidence interval forthe mean mpg for all cars on the road in 2019. Suppose the mpg measurementsfor the population of interest is actually sharply skewed right. For which ofthe sample sizes, \(n=30,50,\) or \(70,\) would the sampling distribution of\(\bar{x}\) be closest to normal? (A) 30 (B) 50 (C) 70 (D) Because of skewness of the population, none of the sampling distributionscan be approximately normal. (E) Because of the central limit theorem, all sampling distributions with \(n\geq 30\) are equally approximately normal.Problem 1Most recent tests and calculations estimate at the \(95 \%\) confidence levelthat mitochondrial Eve, the maternal ancestor to all living humans, lived\(138,000 \pm 18,\) ooo years ago. What is meant by "95\% confidence" in thiscontext? (A) A confidence interval of the true age of mitochondrial Eve has beencalculated using \(z\) -scores of ±1.96 . (B) A confidence interval of the true age of mitochondrial Eve has beencalculated using \(t\) -scores consistent with \(d f\) \(=n-1\) and tailprobabilities of \(\pm 0.025 .\) (C) There is a 0.95 probability that mitochondrial Eve lived between 120,000and 156,000 years ago. (D) If 20 random samples of data are obtained by this method and a \(95 \%\)confidence interval is calculated from each, the true age of mitochondrial Evewill be in 19 of these intervals. (E) Of all random samples of data obtained by this method, \(95 \%\) will yieldintervals that capture the true age of mitochondrial Eve.Problem 2A confidence interval estimate is determined from the GPAs of a simple random sample of \(n\) students. All other things being equal, whichof the following will result in a smaller margin of error? (A) A smaller confidence level (B) A larger sample standard deviation (C) A smaller sample size (D) A larger population size (E) A smaller sample meanProblem 2Suppose (25,30) is a \(90 \%\) confidence interval estimate for a populationmean \(\mu\). Which of the following is a true statement? (A) There is a 0.90 probability that \(\bar{x}\) is between 25 and \(30 .\) (B) Of the sample values, \(90 \%\) are between 25 and 30 . (C) There is a o.9o probability that \(\mu\) is between 25 and 30 . (D) If 100 random samples of the given size are picked and a \(90 \%\)confidence interval estimate is calculated from each, \(\mu\) will be in 90 ofthe resulting intervals. (E) If \(90 \%\) confidence intervals are calculated from all possible samplesof the given size, \(\mu\) will be in \(90 \%\) of these intervals.Problem 3One gallon of gasoline is put into each of an SRS of 30 test autos, and theresulting mileage figures are tabulated with \(\bar{x}\) \(=28.5\) and \(s=1.2 .\)Determine a \(95 \%\) confidence interval estimate of the mean mileage of allcomparable autos. (A) \(28.5 \pm 2.045(1.2)\) (B) \(28.5 \pm 2.045\left(\frac{1.2}{\sqrt{29}}\right)\) (C) \(28.5 \pm 2.045\left(\frac{1.2}{\sqrt{29}}\right)\) (D) \(28.5 \pm 1.96\left(\frac{1.2}{\sqrt{29}}\right)\) (E) \(28.5 \pm 1.96\left(\frac{1.2}{\sqrt{29}}\right)\)Problem 4What sample size should be chosen to find the mean number of absences permonth for school children to within ±0.2 at a \(95 \%\) confidence level if it is known that the standard deviation is \(1.1 ?\) (A) \(n \leq \sqrt{\frac{0.2}{1.96 \times 1.1}}\) (B) \(n \leq \sqrt{\frac{0.2}{1.96 \times 1.1}}\) (C) \(n \leq\left(\frac{1.96 \times 1.1}{0.2}\right)^{2}\) (D) \(n \leq \sqrt{\frac{0.2}{1.96 \times 1.1}}\) (E) \(n \leq\left(\frac{1.96 \times 1.1}{0.2}\right)^{2}\)Problem 5Hospital administrators wish to learn the average length of stay of allsurgical patients. A statistician determines that, for a \(95 \%\) confidencelevel estimate of the average length of stay to within ±0.5 days, 50 surgicalpatients' records will have to be examined. How many records should be lookedat to obtain a \(95 \%\) confidence level estimate to within ±0.25 days? (A) 25 (B) 50 (C) 100 (D) 150 (E) 200Problem 5In a study aimed at reducing developmental problems in lowbirth-weight babies(under 2500 grams), 347 infants were exposed to a special educationalcurriculum while 561 did not receive any special help. After 3 years, thechildren exposed to the special curriculum showed a mean IQ of 93.5 with a standard deviationof 19.1; the other children had a mean IQ of 84.5 with a standard deviation of \(19.9 .\) Find a \(95 \%\) confidenceinterval estimate for the difference in mean IQs of all low-birth-weightbabies who receive special intervention and those who do not. (A) \((93.5-84.5) \pm 1.97\sqrt{\frac{(19.1)^{2}}{347}+\frac{(19.9)^{2}}{561}}\) (B) \((93.5-84.5) \pm1.97\left(\frac{19.1}{\sqrt{307}}+\frac{19.9}{\sqrt{561}}\right)\) (C) \((93.5-84.5) \pm 1.97\sqrt{\frac{(19.1)^{2}}{347}+\frac{(19.9)^{2}}{561}}\) (D) \((93.5-84.5) \pm1.65\left(\frac{19.1}{\sqrt{37}}+\frac{19.9}{\sqrt{561}}\right)\) (E) \((93.5-84.5) \pm 1.65 \sqrt{\frac{(19.1)^{2}+(19.9)^{2}}{347+561}}\)Problem 6Does socioeconomic status relate to age at the time of HIV infection? For 274high-income HIV-positive individuals, the average age of infection was 33.0years with a standard deviation of 6.3 , while for 90 low-income individuals,the average age was 28.6 years with a standard deviation of 6.3 (The Lancet,October 22, 1994, page 1121). Find a \(90 \%\) confidence interval estimate forthe difference in ages of all high- and low-income people at the time of HIVinfection. (A) \(4.4 \pm 0.963\) (B) \(4.4 \pm 1.27\) (C) \(4.4 \pm 2.51\) (D) \(30.8 \pm 2.51\) (E) \(30.8 \pm 6.3\)Problem 7A company manufactures a synthetic rubber bungee cord with a braided coveringof natural rubber and a minimum breaking strength of \(450 \mathrm{~kg}\). Ifthe mean breaking strength of a sample drops below a specified level, theproduction process is halted and the machinery inspected. Which of thefollowing would result from a Type I error? (A) Halting the production process when too many cords break (B) Halting the production process when the breaking strength is below thespecified level (C) Halting the production process when the breaking strength is withinspecifications (D) Allowing the production process to continue when the breaking strength isbelow specifications (E) Allowing the production process to continue when the breaking strength iswithin specifications
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