Student: Date: Instructor: Doug Ensley Course: MAT117 01 Applied Statistics - Ensley Assignment: Online 10 - Section 7.1 1. Consider a sampling distribution with p = 0.12 and samples of size n each. Using the appropriate formulas, find the mean and the standard deviation of the sampling distribution of the sample proportion. a. For a random sample of size n = 5000. b. For a random sample of size n = 1000. c. For a random sample of size n = 500. a. The mean is. b. The mean is. c. The mean is. Summarize the effect of the sample size on the size of the standard deviation. A. As the sample size gets larger, the standard deviation gets smaller. B. As the sample size gets smaller, the standard deviation gets smaller. C. Standard deviation does not depend on the size of the sample. 1 of 5 9/30/15, 4:50 PM
2. For the population of people who suffer occasionally from migraine headaches, suppose p = 0.40 is the proportion who get some relief from taking a certain medicine. For a particular subject, let x = 1 if they get relief and x = 0 if they do not. For a random sample of 38 people who suffer from migraines, answer the following questions. a. State the probability distribution for each observation. Choose the sentence below that best describes the probability distribution. A. For each random sample of 38 people, about 15 get some relief from taking the medicine and 23 do not. B. For each observation, the probability that the medicine helps is 0.40 and the probability that it does not help is 0.60. C. For each random sample of 38 people, about 23 get some relief from taking the medicine and 15 do not. D. For each observation, the probability that the medicine helps is 0.60 and the probability that it does not help is 0.40. b. Find the mean of the sampling distribution of the sample proportion who get relief. mean = (Round to two decimal places as needed.) c. Find the standard deviation of the sampling distribution of the sample proportion. standard deviation = (Round to four decimal places as needed.) d. Explain what the standard deviation in part (c) describes. Choose the sentence below that best describes the standard deviation. A. the standard deviation of the population distribution B. the standard deviation of the sampling distribution C. the probability that the sample mean equals the population mean D. the difference between the sample mean and the population mean 3. In an exit poll, suppose that the mean of the sampling distribution of the proportion of the 3130 people in the sample who voted for recall was 0.65 and the standard deviation was 0.0085. Answer the following questions. a. Based on the approximate normality of the sampling distribution, give an interval of values within which the sample proportion will almost certainly fall. Choose the correct interval below. A. 0.642 to 0.659 B. 0.009 to 0.009 C. 0.026 to 0.026 D. 0.625 to 0.676 b. Based on the result in (a), if you take an exit poll and observe a sample proportion of 0.66, would this be a rather unusual result? Why? A. No, because the observed sample proportion lies outside the interval. B. Yes, because the observed sample proportion lies outside the interval. C. Yes, because the observed sample proportion lies within the interval. D. No, because the observed sample proportion lies within the interval. 2 of 5 9/30/15, 4:50 PM
4. A baseball player in the major leagues who plays regularly will have about 600 at bats (that is, about 600 times he can be the hitter in a game) during a season. Suppose a player has a 0.338 probability of getting a hit in an at-bat. His batting average at the end of the season is the number of hits divided by the number of at-bats. This batting average is a sample proportion, so it has a sampling distribution describing where it is likely to fall. Complete parts a and b below. a. Describe the shape, mean, and standard deviation of the sampling distribution of the player's batting average after a season of 600 at-bats. Describe the shape of the sampling distribution. Choose the correct response below. A. The distribution is bell-shaped, centered on a standard deviation of 0.019, and has a minimum value of 0 and a maximum value of one mean value. B. The distribution is bell-shaped, centered on a standard deviation of 0.019, and the majority of the distribution lies within three mean values of the standard deviation. C. The distribution is bell-shaped, centered on a mean of 0.338, and the majority of the distribution lies within three standard deviations of the mean. D. The distribution is bell-shaped, centered on a mean of 0.338, and has a minimum value of 0 and a maximum value of one standard deviation. b. Explain why a batting average of 0.319 or 0.357 would not be especially unusual for this player's year-end batting average. A. Year-end batting averages of 0.357 and 0.319 lie one standard deviation from the mean. Therefore, it is not unlikely that a player with a career batting average of 0.338 would have a year-end batting average of 0.357 or 0.319. B. Year-end batting averages of 0.357 and 0.319 lie three standard deviations from the mean. Therefore it is almost a certainty that the player will recieve a year-end batting average between 0.357 and 0.319. C. Year-end batting averages of 0.357 and 0.319 lie three standard deviations from the mean. Therefore it is extremely unlikely that the player will recieve a year-end batting average of 0.357 or 0.319. 3 of 5 9/30/15, 4:50 PM
5. The figure illustrates two sampling distributions for sample proportions when the population proportion p = 0.55. Complete parts a through c. n = 2000 n = 200 0.4 0.5 0.6 0.7 x a. Find the standard deviation for the sampling distribution of the sample proportion with (i) n = 200, (ii) n = 2000. (i) standard deviation = (ii) standard deviation = (Round to four decimal places as needed.) b. Explain why the sample proportion would be very likely to fall (i) between 0.44 and 0.66 when n = 200, and (ii) between 0.52 and 0.58 when n = 2000. A. The sample proportion has to be between 0 and 1. B. The sample proportion is very likely to equal the population proportion when n is larger. C. The sample proportion has to be close to the population proportion. D. The sample proportion is very likely to fall within three standard deviations of the mean. c. Explain how the results in (b) indicate that the sample proportion is closer to the population proportion when the sample size is larger. A. When n is larger, the standard deviation is smaller, so the interval is larger. B. When n is larger, the standard deviation is smaller, so the interval is smaller. C. When n is larger, the standard deviation is larger, so the interval is smaller. D. When n is larger, the standard deviation is larger, so the interval is larger. 4 of 5 9/30/15, 4:50 PM
1. 0.12 0.0046 0.12 0.0103 0.12 0.0145 A. As the sample size gets larger, the standard deviation gets smaller. 2. B. For each observation, the probability that the medicine helps is 0.40 and the probability that it does not help is 0.60. 0.40 0.0795 B. the standard deviation of the sampling distribution 3. D. 0.625 to 0.676 D. No, because the observed sample proportion lies within the interval. 4. C. The distribution is bell-shaped, centered on a mean of 0.338, and the majority of the distribution lies within three standard deviations of the mean. A. Year-end batting averages of 0.357 and 0.319 lie one standard deviation from the mean. Therefore, it is not unlikely that a player with a career batting average of 0.338 would have a year-end batting average of 0.357 or 0.319. 5. 0.0352 0.0111 D. The sample proportion is very likely to fall within three standard deviations of the mean. B. When n is larger, the standard deviation is smaller, so the interval is smaller. 5 of 5 9/30/15, 4:50 PM