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Name: __________________________ Date: _____________ 1. A medical researcher is working on a new treatment for a certain type of cancer. The average survival time after diagnosis on the standard treatment is two years. In an early trial, she tries the new treatment on three subjects who have an average survival time after diagnosis of four years. Although the survival time has doubled, the results are not statistically significant, even at the 0.10 significance level. Suppose, in fact, that the new treatment does increase the mean survival time in the population of all patients with this particular type of cancer. Which of the following statements is true? A) A type I error has been committed. B) A type II error has been committed. C) No error has been committed. 2. A sprinkler system is being installed in a large office complex. Based on a series of test runs, a 99% confidence interval for , the average activation time of the sprinkler system (in seconds), is found to be (22, 28). Determine whether each of the following statements is true or false. A) The 99% confidence level implies that P(22 < < 28) = 0.99. B) The 99% confidence level implies that P(22 < x < 28) = 0.99. C) The 99% confidence level implies that 99% of the sample means ( x ) obtained from repeated sampling would fall between 22 and 28. D) If a 95% confidence interval were calculated from the same data, (23, 27) would be a possible interval. 3. A simple random sample of 100 athletes is selected from a large high school. In the sample, there are 15 football players. What is the standard error of the sample proportion of football players? A) 0.00128 B) 0.0357 C) 0.05 D) 0.357 4. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin of error for a confidence interval for the mean , based on a specified sample size n, increases as the confidence level decreases. C) The margin of error for a 95% confidence interval for the mean decreases as the population standard deviation decreases. D) The sample size required to obtain a confidence interval of specified margin of error m, increases as the confidence level increases. Page 1 5. The test statistic for a two-sided significance test for a population mean is z = –2.12. What is the corresponding P-value? A) 0.017 B) 0.034 C) 0.483 D) 0.983 6. A simple random sample of 85 students is taken from a large university on the West Coast to estimate the proportion of students whose parents bought a car for them when they left for college. When interviewed, 51 students in the sample responded that their parents bought them a car. What is a 95% confidence interval for p, the population proportion of students whose parents bought a car for them when they left for college? A) (0.296, 0.504) B) (0.463, 0.737) C) (0.496, 0.704) D) (0.513, 0.687) Use the following to answer questions 7-9: The distribution of the amount of money undergraduate students spend on books for a term is slightly right skewed, with a mean of $400 and a standard deviation of $80. 7. If a student is selected at random, what is the probability that this student spends more than $425 on books? A) 0.1125 B) 0.3773 C) 0.6227 D) This cannot be determined from the information given. 8. If a simple random sample of 100 undergraduate students is selected, what is the probability that these students spend more than $425 on books, on average? A) 0.00089 B) 0.2353 C) 0.3773 D) This cannot be determined from the information given. Page 2 9. In a simple random sample of 100 undergraduate students, what is the expected value of the sample mean amount of money spent on books? A) $400 B) Anywhere between $320 and $480. C) Anywhere between $392 and $408. D) This cannot be determined from the information given. Use the following to answer questions 10-12: During the last student elections at a certain college, 45% of the students voted for the democratic student party. A simple random sample of students from this college is to be selected. 10. If 12 students are to be selected, what is the distribution of the number of students in the sample who voted for the democratic student party? 11. If 12 students are to be selected, what is the probability that more than 7 students in the sample voted for the democratic student party? 12. If 120 students are to be selected, what is the (approximate) distribution of the number of students in the sample who voted for the democratic student party? Use the following to answer questions 13-14: A simple random sample of 25 male faculty members at a large university reveals that 10 of them feel that the university is supportive of female and minority faculty. An independent simple random sample of 20 female faculty members reveals that only 5 of them feel that the university is supportive of female and minority faculty. Let p1 and p2 represent the proportions of all male and female faculty members at the university who feel that the university is supportive of female and minority faculty at the time of the survey. 13. Is there evidence that the proportion of male faculty members who feel the university is supportive of female and minority faculty is larger than the corresponding proportion of female faculty members? To determine this, test the hypotheses H0: p1 = p2 versus Ha: p1 > p2. What can we say about the P-value of the hypothesis test? A) The P-value is smaller than 0.001. B) The P-value is between 0.001 and 0.01. C) The P-value is between 0.01 and 0.05. D) The P-value is larger than 0.05. Page 3 14. What is the margin of error for a 95% plus four confidence interval for p1 – p2? A) m = 0.134 B) m = 0.220 C) m = 0.263 D) m = 0.345 15. An engineer has designed an improved light bulb. The previous design had an average lifetime of 1200 hours. Based on a sample of 2000 of these new bulbs, the average lifetime was found to be 1201 hours. Although the difference is quite small, the effect was statistically significant. What is the best explanation? A) New designs typically have more variability than standard designs. B) The sample size is very large, so that even a small difference can be detected. C) The mean of 1200 is large. D) The relative improvement in average lifetime is 0.000083, which is much smaller than 0.05. Use the following to answer questions 16-17: Assume that sample data, based on two independent samples of size 25, gives us x1 = 505, x2 = 515, s1 = 23, and s2 = 28. 16. What is a 95% confidence interval (use the conservative value for the degrees of freedom) for 2 – 1? A) (–2.40, 22.40) B) (–4.96, 24.96) C) (–4.57, 24.57) D) (5.79, 14.21) 17. Determine whether each of the following statements is true or false. A) Based on the confidence interval, we can conclude, at the 5% significance level, that there is no difference between the two population means, 2 and 1. B) The margin of error for the difference between the two sample means would be smaller if we were to take larger samples. C) If we were to use the unpooled t test with the more accurate approximation for the degrees of freedom (used in software), the degrees of freedom would be 51. D) If a 99% confidence interval were calculated instead of the 95% interval, it would include more values for the difference between the two population means. Page 4 18. The square footage of the several thousand apartments in a new development is advertised to be 1250 square feet, on average. A tenant group thinks that the apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicions. Let represent the true average area (in square feet) of these apartments. What are the appropriate null and alternative hypotheses? A) H0: = 1250 vs. Ha: < 1250 B) H0: = 1250 vs. Ha: 1250 C) H0: = 1250 vs. Ha: > 1250 19. A simple random sample of 60 blood donors is taken to estimate the proportion of donors with type A blood with a 95% confidence interval. In the sample, there are 10 people with type A blood. What is the margin of error for this confidence interval? A) 0.048 B) 0.079 C) 0.094 D) 1.96 20. The test statistic for a significance test for a population mean is z = –2.12. The hypotheses are H0: = 10 versus Ha: > 10. What is the corresponding P-value? A) 0.017 B) 0.034 C) 0.483 D) 0.983 Use the following to answer questions 21-23: Chromosome defect A occurs in only one out of 200 adult males. A random sample of 100 adult males is selected. Let the random variable X represent the number of males in the sample who have this chromosome defect. 21. What is the exact distribution of the random variable X? 22. Can we use the normal approximation to answer probability questions about the random variable X? Briefly explain why or why not. 23. What are the mean and standard deviation of the random variable X? Page 5 Use the following to answer questions 24-25: In the university library elevator there is a sign indicating a 16-person limit as well as a weight limit of 2500 lbs. Suppose that weight of students, faculty, and staff is approximately normally distributed with a mean weight of 150 lbs. and a standard deviation of 27 lbs. 24. What is the probability that the random sample of 16 people in the elevator will exceed the weight limit? 25. When the elevator is full, we can think of the 16 people in the elevator as a simple random sample of people on campus. What average weight for these 16 people in the elevator will result in the total weight exceeding the weight limit of 2500 lbs.? 26. The tail area above a test statistic value of z = 1.812 is 0.035. Determine whether each of the following statements is true or false. A) If the alternative hypothesis is of the form Ha: > 0, the data are statistically significant at significance level = 0.05. B) If the alternative hypothesis is of the form Ha: > 0, the data are statistically significant at significance level = 0.10. C) If the alternative hypothesis is of the form Ha: 0, the data are statistically significant at significance level = 0.05. D) If the alternative hypothesis is of the form Ha: 0, the data are statistically significant at significance level = 0.10. 27. Ten years ago, at a small high school in Alabama, the mean Math SAT score of all high school students who took the exam was 490 with a standard deviation of 80. This year, the Math SAT scores of a random sample of 25 students who took the exam are obtained. The mean score of these 25 students is x = 525. To determine if there is evidence that the scores in the district have improved, the hypotheses H0: = 490 versus Ha: > 490 are tested at the 5% significance level. The P-value is found to be 0.014. Suppose that the average Math SAT score of all high school students at this high school is in fact equal to 505. Which of the following statements is true? A) A type I error has been committed. B) A type II error has been committed. C) No error has been committed. Page 6 28. In a test of statistical hypotheses, what does the P-value tell us? A) If the null hypothesis is true. B) If the alternative hypothesis is true. C) The largest level of significance at which the null hypothesis can be rejected. D) The smallest level of significance at which the null hypothesis can be rejected. Use the following to answer questions 29-31: A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district of a large city. The researcher obtained a simple random sample of 60 high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be 6 hours with a standard deviation of 3 hours. The researcher also obtained an independent simple random sample of 40 high school students in a large city school district and found the mean time spent in extracurricular activities per week to be 4 hours with a standard deviation of 2 hours. Let 1 and 2 represent the mean amount of time spent in extracurricular activities per week by the populations of all high school students in the suburban and city school districts, respectively. Assume two sample t procedures are safe to use. 29. What is a 95% confidence interval for 1 – 2? (Use the conservative value for the degrees of freedom.) A) 2 ± 0.5 hours B) 2 ± 0.84 hours C) 2 ± 1.01 hours D) 2 ± 1.34 hours 30. Suppose the researcher had wished to test the hypotheses H0: 1 = 2 versus Ha: 1 2. What can we say about the value of the P-value? A) P-value < 0.01 B) 0.01 < P-value < 0.05 C) 0.05 < P-value < 0.10 D) P-value > 0.10 31. If we had used the more accurate software approximation to the degrees of freedom, what would be the number of degrees of freedom for the two sample t procedures? A) 39 B) 59 C) 97.998 D) 99.286 Page 7 32. Is the mean height for all adult American males between the ages of 18 and 21 now over 6 feet? Let represent the population mean height of all adult American males between the ages of 18 and 21. What are the appropriate null and alternative hypotheses to answer this question? A) H0: = 6 vs. Ha: < 6 B) H0: = 6 vs. Ha: 6 C) H0: = 6 vs. Ha: > 6 Use the following to answer questions 33-34: Do students tend to improve their SAT Mathematics (SAT-M) score the second time they take the test? A random sample of four students who took the test twice received the following scores. Student 1 2 3 4 First score 450 520 720 600 Second score 440 600 720 630 Assume that the change in SAT-M score (second score – first score) for the population of all students taking the test twice is normally distributed. 33. Suppose we do not believe that students tend to improve their SAT-M score the second time they take the test. Based on the confidence interval previously calculated, we wish to test H0: = 0 versus Ha: 0 at the 5% significance level. Determine which of the following statements is true: A) We cannot make a decision since the confidence interval is so wide. B) We cannot make a decision since the confidence level we used to calculate the confidence interval is 90%, and we would need a 95% confidence interval. C) We accept H0, since the value 0 falls in the 90% confidence interval and would therefore also fall in the 95% confidence interval. D) We reject H0, since the value 0 falls in the 90% confidence interval. 34. What is a 90% confidence interval for , the mean change in SAT-M score? A) (–8.24, 58.24) B) (–18.08, 68.08) C) (–22.56, 72.56) D) (–39.31, 89.31) Page 8 Use the following to answer questions 35-37: A simple random sample of 60 households in the city of Greenville (call this city 1) is taken. In the sample, there are 45 households that decorate their houses with lights for the holidays. A simple random sample of 50 households is also taken from the neighboring town of Brownsboro (call this city 2). In the sample, there are 40 households that decorate their houses. We wish to estimate the difference in proportions of households that decorate their houses with lights for the holidays with a 95% confidence interval. 35. Under the null hypothesis of equality of the proportions of households that decorate their houses in the two neighboring towns, what is the standard error of the difference in sample proportions? A) 0.040 B) 0.0795 C) 0.0802 D) 0.1125 36. What is a 95% confidence interval for the difference in population proportions of households that decorate their houses with lights for the holidays? A) (–0.181, 0.081) B) (–0.206, 0.106) C) (–0.231, 0.138) D) (–0.255, 0.155) 37. What is the standard error of the difference in sample proportions? A) 0.040 B) 0.0795 C) 0.0802 D) 0.1125 Page 9 38. A small New England college has a total of 400 students. The Math SAT is required for admission, and the mean score of all 400 students is 620. The population standard deviation is found to be 60. The formula for a 95% confidence interval yields the interval 640 ± 5.88. Determine whether each of the following statements is true or false. A) If we repeated this procedure many, many times, only 5% of the 95% confidence intervals would fail to include the mean Math SAT score of the population of all students at this college. B) The probability that the population mean will fall between 634.12 and 645.88 is 0.95. C) The interval is incorrect. It is much too narrow. D) If we repeated this procedure many, many times, x would fall between 634.12 and 645.88 about 95% of the time. Use the following to answer question 39: Central Middle School has calculated a 95% confidence interval for the mean height () of 11-year old boys at their school and found it to be 56 ± 2 inches. 39. Determine whether each of the following statements is true or false. A) There is a 95% probability that is between 54 and 58. B) There is a 95% probability that the true mean is 56, and there is a 95% chance that the true margin of error is 2. C) If we took many additional random samples of the same size and from each computed a 95% confidence interval for , approximately 95% of these intervals would contain . D) If we took many additional random samples of the same size and from each computed a 95% confidence interval for , approximately 95% of the time would fall between 54 and 58. 40. An engineer has designed an improved light bulb. The previous design had an average lifetime of 1200 hours. Using a sample of 2000 of these new bulbs, the average lifetime of this improved light bulb is found to be 1201 hours. Although the difference is quite small, the effect was statistically significant at the 0.05 level. Suppose that, in fact, there is no difference between the mean lifetimes of the previous design and the new design. Which of the following statements is true? A) A type I error has been committed. B) A type II error has been committed. C) No error has been committed. Page 10 Use the following to answer questions 41-44: Your company is producing special battery packs for the most popular toy during the holiday season. The life span of the battery pack is known to be normally distributed with a mean of 250 hours and a standard deviation of 20 hours. 41. What would typically be a better distribution than the normal distribution to model the lifespan of these battery packs? 42. If a simple random sample of four battery packs is selected from your company and we assume that their lifespans are independent, what is the probability that they all last longer than 260 hours? 43. If a simple random sample of four battery packs is selected from your company, what is the probability that the average lifetime of these four packs is longer than 260 hours? 44. What percentage of battery packs lasts longer than 260 hours? Page 11 Answer Key 1. B 2. A) False, B) False, C) False, D) True 3. B 4. A) False, B) False, C) True, D) True 5. B 6. C 7. D 8. A 9. A 10. B(12, 0.45) 11. 0.1117 12. N(54, 5.45) 13. D 14. C 15. B 16. C 17. A) True, B) True, C) False, D) True 18. A 19. C 20. D 21. B(100, 0.005) 22. No, np = (1/200)(100) = 0.5. The normal approximation can be used when np 10 and n(1 – p) 10. 23. X np (100)(0.005) 0.5 , X np(1 p) 100(0.005)(0.995) 0.705 24. 0.1772 25. 156.25 lbs. 26. A) True, B) True, C) False, D) True 27. C 28. D 29. C 30. A 31. C 32. C 33. C 34. C 35. C 36. B 37. B 38. A) False, B) False, C) False, D) False 39. A) False, B) False, C) True, D) False 40. A 41. The Weibull distribution. 42. 0.00906 43. 0.1587 Page 12 44. 30.85% Page 13

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