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Solve the following problems: Problem 1

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Solve the following problems: Problem 1. Although a machine is designed to produce tennis balls with an average diameter of 3 inches, the manufacturer is concerned that the diameters no longer conform to this specification. However, before shutting down production, the manufacturer wants to be quite sure that the mean diameter differs from 3 inches. Therefore, a hypothesis test is to be performed. a. Determine the null hypothesis. (3 points) b. Determine the alternative hypothesis. (3 points) c. Classify the hypothesis test as two-tailed, left-tailed or right-tailed. (3 points) Problem 2. In exercises 9.63, 9.64 and 9.66, we have given the value obtained for the test statistic z in a one-mean z test. We have also specified whether the test is two-tailed left-tailed or righttailed. Determine the p-value in each case and decide whether, at the 5% significance level, the data provides sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. (1.5 points each – 9 points total) 9.63 Right tailed test: a. z = 2.03 b. z = -0.31 9.64 Left-tailed test: a. z = -1.84 b. z = 1.25 9.66 two-tailed test: a. z = 3.08 b. z =-2.42 Problem 3. A market researcher wants to perform a test with the intent of establishing that his company’s pump bottle of soap has a mean lifetime greater than 40 days. The sample size is 70 and the researcher knows that the standard deviation σ=5.6. Assume that the lifetimes of pump bottles of soap follow a normal distribution. a. If you set the rejection region to be R: ?? ≥ 41.31, what is the level of significance of your test? (4 points) b. Find the numerical value of c such that the test R: ?? ≥ ? has a 5% level of significance. (4 points) Problem 4. Researchers induced the growth of tumors in a sample of 9 mice. The average size of the tumors in the sample was ?? = 4.3 mm and the sample standard deviation was s = 1.2 mm. Test whether the mean tumor size differs from 4.0 mm. Use α=.10 a. State the null and alternative hypotheses. b. Determine the rejection region – indicate how you did this; don’t just give the answer. c. Compute the test statistic – indicate what formula was used. d. What is your conclusion and why? Problem 5. To compare two programs for training industrial workers to perform a skilled job, 20 workers are included in an experiment. Of these, 10 are selected at random and trained by method 1; the remaining 10 are trained by method 2. After completion of training, all the workers take a time-and-motion test that records the speed of performance of the skilled job. The following data are obtained: Method 1 Method 2 15 20 23 31 11 23 13 19 16 21 23 17 18 16 28 26 27 24 25 28 Construct a 95% confidence interval for the mean difference in job times between the two methods. Assume the true variances of job times are the same for the two methods. Problem 6. Researchers at Carleton University in Canada studied a random sample of 10 first-year students who lived on campus and a random sample of 10 first-year students who lived off campus. They measured the weight gain of these students (in kg) during their first year of college. Their study yielded the following summary statistics: On-campus Off-campus ???1 = 2.46 ? ???2 = 0.85 ? ?1 = 3.26 ?2 = 2.03 ?1 = 10 ?2 = 10 Obtain a 95% confidence interval for the difference ?1 − ?2 between the mean weight gain of first-year students who lived on-campus and first-year students who lived off-campus. Assume that the standard deviations of these two populations are different. Problem 7. A 2001 article described a study in which sixth-grade students who had not previously played chess participated in a program in which they took chess lessons and played chess daily for 9 months. Each student took a memory test before starting the chess program and again at the end of the 9-month period. Data are given in the table below: Student Pre-test Post-test 1 510 850 2 610 790 3 640 850 4 675 775 5 600 700 6 550 775 7 610 700 8 625 850 9 450 690 10 720 775 11 575 540 12 675 680 At the 5% significance level, do the data provide sufficient evidence to conclude that, on average, students who participated in the chess program have higher memory test scores? (Note: the mean and standard deviation of the paired differences are ?? =– 144.58 and s = 109.74, respectively.) a. Indicate the null and alternative hypotheses. b. What is the rejection region (the set of values of the test statistic for which we can reject the null hypothesis)? c. Compute the test statistic. d. What is your conclusion? Problem 8. Plastic sheets produced by a machine are periodically monitored for possible fluctuations in thickness. Uncontrollable heterogeneity in the viscosity of the liquid mold makes some variation in thickness measurements unavoidable. However, if the true standard deviation of thickness exceeds 1.5 mm, there is cause to be concerned about the product quality. Thickness measurements (in millimeters) of 10 specimens produced on a particular shift are as follows: 226 228 226 225 232 228 227 229 225 230 Construct a 95% confidence interval for the true standard deviation of the thickness of sheets produced on this shift. (Note: from the sample data, s = 2.2706). Problem 9. A newspaper article reported the results of a survey of 1100 drivers. Of those surveyed, 990 admitted to careless or aggressive driving during the previous 6 months. Assuming that it is reasonable to regard this sample of 1100 as representative of the population of drivers, we can use this information to construct a 90% confidence interval for p, the proportion of all drivers who have engaged in careless or aggressive driving in the past 6 months. Problem 10. A public health survey is to be designed to estimate the proportion p of a population having defective vision. How many persons should be examined if the public health commissioner wishes to be 98% certain that the margin of error is below .05 when: a. There is no knowledge about the value of p? b. p is known to be about 0.3? Problem 11. A five-year-old census recorded that 20% of the families in a large community lived below the poverty level. To determine if this percentage has changed, a random sample of 400 families is studied and 70 are found to be living below the poverty level. Does this finding indicate that the current percentage of families earning incomes below the poverty has changed from what it was five years ago? a. Indicate the null and alternative hypotheses. ( b. What is the rejection region (the set of values of the test statistic for which we can reject the null hypothesis)? c. Compute the test statistic. d. What is your conclusion? e. What is the p-value? Problem 12. As part of a study, each person in a sample of 258 cell phone users aged 20 to 39 was asked if they use their cell phones to stay connected while they are in bed; 168 said that they did. The same question was also asked of each person in a sample of 129 cell phone users aged 40 to 49; of these, 61 said that they did. Construct a 90% confidence interval for the difference in population proportions. Stat 211, Section 01 – Spring 2021 FINAL EXAM NAME _______________________________________________________________ ANSWERS TO SHORT ANSWER QUESTIONS 1. __________________________ 2. __________________________ 3. __________________________ 4. __________________________ 5. __________________________ 6. __________________________ 7. __________________________ 8. __________________________ 9. __________________________ 10. __________________________ 11. __________________________ 12. __________________________ 13. ___________________________ Multiple Choice Question 1 Consider random variable X that has a binomial distribution with n = 40 and p = .15. We wish to compute the probability P(X≤3) for this binomial distribution. Assume that either an exact solution or a solution based on a normal approximation would be acceptable. Which of the following would be an acceptable solution? A) (40 ). 150 . 8540 + (40 ). 151 . 8539 + (40 ). 152 . 8538 0 1 2 B) ?(? < 2.5−6 ) ( )( √ 40 0.15)(0.85) 3.5−6 ) √(40)(0.15 )(0.85) C) 1 − ? (? > D) (40 ). 153 . 8537 3 E) None of the above will calculate the desired probability. Multiple Choice Question 2 The probability density function for a uniform distribution on the interval [1, 4] equals: : A) ?(?) = .4 B) ?(?) = .3 C) ?(?) = 4.0 D) ?(?) = 3.0 E) ?(?) = 1/3 Multiple Choice Question 3 Let X be a normally-distributed random variable with mean ? = 55.5 and standard deviation σ =2.7 Find b such that P(X < b) = .90 A) 61.7802 B) 58.956 C) 59.9415 D) 60.7920 E) None of the above Multiple Choice Question 4 Students may choose between a 3-credit Physics course without labs and a 4-credit Physics course with labs. There were 12 students in the section with labs; they had a mean grade of 85 with a standard deviation of 5. There were 20 students in the section without labs. They had a mean grade of 76 with a standard deviation of 6. We intend to compute a confidence interval for the difference in means, and we assume that the true population variances are equal. ? 1 = 12 ? 2 = 20 ??? ?1 = 85 ? ??? ??2 = 76 ?12 = (5) 2 ?2 2 = (6) 2 ?1 − ??2? What is the best estimate of the standard deviation of ? A) B) C) √ √ 2 (5)2 12 √ + 1 32 1 ( 12 + 20 ) (6) 2 20 (12 (5)2+20(6)2 ) 12 D) E) (52 +62 ) 32 20 5+ 32 30 1 6 11 (5) 2 +19(6) 2 √ 1 (12 + 20) 1 1 (12 + 20 ) Multiple Choice Question 5 Which of the following will increase the width of a confidence interval for a proportion? A) B) C) D) E) Reduce the confidence level α Increase the sample size Add .5 to the sample proportion Replace the sample proportion with 0.5 None of these will increase the width of the confidence interval. Multiple Choice Question 6 Beginning students in accounting took a test and a Computer Anxiety Score was assigned to each student. A small population standard deviation would indicate that computer anxiety is nearly the same for all beginning students. For a sample of 15 students, the sample standard deviation was s = 0.7727. Which of the following represents a 90% confidence interval for the population standard deviation? A) √ (0.7727)(15) 23.685 B) 0.7727√ C) 0.7727√ D) 0.7727√ E) √