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SAMPLING DISTRIBUTIONS REVIEW 1. If we double the sample size, then ο³ x will be multiplied by: (a) 2 (b) 1 (d) 1 2 (c) 2 2 (e) 4 2. If a statistic used to estimate a parameter is such that the mean of its sampling distribution is different from the true value of the parameter being estimated, the statistic is said to be a) Random b) Biased . c) A proportion d) Unbiased e) None of the above. The answer is 3. The Gallup Poll has decided to increase the size of its random sample of Canadian voters from about 1500 people to about 4000 people. The effect of this increase is to: a) reduce the bias of the estimate b) increase the standard error of the estimate c) reduce the variability of the estimate d) increase the confidence interval width for the parameter e) have no effect because the population size is the same 4. A random sample of 100 observations is to be drawn from a population with a mean of 40 and a standard deviation of 25. The probability that the mean of the sample will exceed 45 is: a) .4772 b) .4207 c) .0793 d) .0228 e) not possible to compute based on the information provided 5. The sample mean is an unbiased estimator for the population mean. This means a) the sample mean always equals the population mean b) the average sample mean, over all possible samples, equals the population mean c) the sample mean is always very close to the population mean d) the sample mean will only vary a little from the population mean e) the sample mean has a normal distribution 6. A machine is designed to fill 16-ounce bottles of shampoo. When the machine is working properly, the mean amount poured into the bottles is 16.05 ounces with a standard deviation of 0.1 ounce. If four bottles are randomly selected each hour and the number of ounces in each bottle is measured, then approximately 95% of the observations should occur in which interval? a) 16.05 and 16.15 ounces d) 15.90 and 16.20 ounces b) β.30 and +.30 ounces e) c) 15.95 and 16.15 ounces 16 and 16.1 7. A can of salmon has a stated net weight of 250 grams. However, due to the variation in the canning process, the actual net weight has an approximate normal distribution with a mean of 255 grams and a standard deviation of 10 grams. According to Consumer Affairs, a sample of 16 cans should have less than a 5% chance that the mean weight is less than 250 grams. What is the actual probability that a sample of 16 cans will have a mean weight less than 250 grams? a) .1915 b) .3085 c) .0228 d) .4772 e) .0500 8. A phone-in poll conducted by a newspaper reported that 73% of those who called in liked business tycoon Donald Trump. The number 73% is a a) statistics b) sample c) parameter d) population e) size 9. The Central Limit Theorem states that: a) if n is large then the distribution of the sample can be approximated closed by a normal curve b) if n is large, and if the population is normal, then the variance of the sample mean must be small c) if n is large, then the sampling distribution of the sample means can be approximated by a normal model d) if n is large, and if the population is normal, then the sampling distribution of the sample means can be approximated closely by a normal curve e) if n is large, then the variance of the sample must be small 10. Which of the following statements regarding the sampling distribution of sample means is incorrect? a) The sampling distribution is approximately normal when sampling from a normal population or the sample size is sufficiently large. b) The mean of the sampling distribution is the mean of the population. c) The standard deviation of the sampling distribution is the standard deviation of the population. d) The sampling distribution is found by taking repeated samples of the same size from the population of interest and computing the mean of all possible samples. e) All of these are correct. For questions 11-13: A survey asks a random sample of 1500 adults in San Antonio area if they support an increase in the sales tax from 8.25% to 9% with the additional revenue going to education (teachersβ bonus checks at Thanksgiving timeβ¦to buy a real turkey). Suppose 30% of all San Antonio adults support the increase. 11. The mean of the sampling distribution of πΜ is a) 8.25% b) 30% ο± 8.25% c) 0.30 d) 1500 e) 450 12. The standard deviation of the sampling distribution of pΜ is a) 0.4582 b) 0.2100 c) 0.0118 d) 0.0141 e) 0.0000 13. The probability that pΜ is more than 0.40 is a) less than 0.0001 b) about 0.100 d) 0.50 e) 0.8918 c) 0.4549 14. After repeated observations, it has been determined that the waiting time at the drive-through window at a local bank on Friday afternoons between 12:00 noon and 6:00 pm is skewed left with a mean of 3.5 minutes and standard deviation of 1.9 minutes. A sample of 100 customers is to be taken next Friday. What is the probability that the mean of the sample will exceed 4 minutes? a) 0.0042 b) 0 .0396 c) 0 .0420 d) 0.3960 e) The probability cannot be determined using a normal curve For questions 15-19: The distribution of actual weights of chocolate bars produced by a certain machine is normal with mean 8.1 ounces and standard deviation of 0.1 ounces. 15. If a sample of five of these chocolate bars is selected, the probability that their average weight is less than 8 ounces is a) 0.0127 b) 0.1853 c) 0.4871 d) 0.9873 e) Not enough information provided 16. If a sample of five of these chocolate bars is selected, there is only 5% chance that the average weight of the sample of five of the chocolate bars will be below a) 7.94 ounces b) 8.03 ounces c) 8.08 ounces d) 8.20 ounces e) 8.29 ounces 17. The company sells the individually (& independently selected) wrapped eight chocolate bars as a packaged special product, what is the expected weight of the special packaged product (chocolate only)? a) 0.8 ounces b) 8.1 ounces c) 40.5 ounces d) 64.8 ounces e) cannot be determined. 18. What is the standard deviation of the sampling distribution of the weight of the special packaged product of chocolate only? a) 0.01 ounces b) 0.08 ounces c) 0.2828 ounces d) 0.8944 ounces e) cannot be determined. 19. What is the probability that a special packaged product contains more than 66 ounces? a) 0.0000 b) 0.0010 c) 0.1100 d) 1.0000 e) cannot be determined. 20. Which of the following statements best describes a sampling distribution of sample means: a) It is π₯Μ . b) It is the distribution of all possible values of a population parameter. c) It is the distribution of values of a statistic taken from all possible samples of a specific size. d) It is an unbiased estimator π. e) It is the normal distribution with π₯Μ = 0 πππ π = 1. 21. A sample of 5000 female adults was randomly drawn from the United States. It is known that the diastolic blood pressure for adult women in the United States is N(80, 12). What is the mean and standard deviation of the distribution of the sample means? a) ππ₯Μ = .16 , ππ₯Μ = .1697 b) ππ₯Μ = 80 , ππ₯Μ = 12 c) ππ₯Μ = 80 , ππ₯Μ = .1697 d) ππ₯Μ = 3.58 , ππ₯Μ = .024 e) Cannot be determined from the information given. 22. Which of the following statements are true? I. The sampling distribution of πΜ has a mean equal to the population proportion p. II. The sampling distribution of πΜ has a standard deviation equal to βπ π (1 β π) III. The sampling distribution of πΜ is considered close to normal provided that π β₯ 30. a) I b) I and II c) II and III d) I, II, and III e) I and III 23. Suppose that 35% of all business executives are willing to switch companies if offered a higher salary. If a headhunter randomly contacts an SRS of 100 executives, what is the probability that over 40% will be willing to switch companies if offered a higher salary? a) .1469 b) .1977 c) .4207 d) .8023 e) .8531 24. The average outstanding bill for delinquent customer accounts for a national department store chain is $187.50 with a standard deviation of $54.50. In a simple random sample of 50 delinquent accounts, what is the probability that the mean outstanding bill is over $200? a) .0526 b) .0667 c) .4090 d) .5910 e) .9474 25. The average number of daily emergency room admissions at a hospital is 85 with a standard deviation of 37. In a simple random sample of 30 days, what is the probability that the mean number of daily emergency admissions is between 75 and 95? a) .1388 b) .2128 c) .8612 d) .8990 e) .9970 Questions 26-29: Information on a packet of seeds claims that the germination rate is 92%. Each packet contains 160 seeds. 26. If πΜ is the proportion of the seeds that germinate, what is the mean of the sampling distribution of Μπ? 27.What is the standard deviation of the sampling distribution of πΜ ? 28. Check that you can use the normal approximation for the distribution of πΜ . 29. What is the probability that more than 95% of the 160 seeds in the packet will germinate? 30-33 A certain beverage company is suspected of underfilling its cans of soft drink. The company advertises that its cans contain, on the average, 12 ounces of soda with standard deviation 0.4 ounce. 30. Compute the probability that a random sample of 50 cans produces a sample mean fill of 11.9 ounces or less. 31. Suppose all of the possible samples of 50 cans were taken and the mean number of ounces of soda calculated. Describe the approximate shape of the distribution for these values of x . 32. What important principle that we studied is used to answer the previous question? 33. Can you calculate the probability that a single randomly selected can contains 11.9 ounces or less? If so, do it. If not, explain why you cannot.