Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let μ denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (7.6, 9.1). (a) Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning. O The 90% would be the same since the z critical value for 90% is the same as the z critical value for 95%. The 90% would be narrower since the z critical value for 90% is larger than the z critical value for 95%. The 90% would be wider since the z critical value for 90% is larger than the z critical value for 95%. O The 90% would be narrower since the z critical value for 90% is smaller than the z critical value for 95%. O The 90% would be wider since the z critical value for 90% is smaller than the z critical value for 95%. (b) Consider the following statement: There is a 95% chance that is between 7.6 and 9.1. Is this statement correct? Why or why not? O It is not a correct statement. We are 95% confident in the general procedure for creating the interval, but the mean may or may not be enclosed in this interval. O It is not a correct statement. Each interval contains the mean by definition. O It is a correct statement. Each interval contains the mean by definition. O It is a correct statement. There is only a 5% chance that the mean is not between these values. It is not a correct statement. There is a 5% chance that the mean is between these values. (c) Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.6 and 9.1. Is this statement correct? Why or why not? O It is not a correct statement. The interval is an estimate for the sample mean, not a boundary for sample values. It is not a correct statement. The interval is an estimate for the sample mean, not a boundary for population values. O It is a correct statement. This interval is a great estimate of boundaries for population values. O It is a correct statement. This is the definition of a confidence interval. It is not a correct statement. The interval is an estimate for the population mean, not a boundary for population values. (d) Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% interval is repeated 100 times, 95 of the resulting intervals will include u. Is this statement correct? Why or why not? O It is not a correct statement. We expect 5 out of the 100 intervals to contain the mean. O It is not a correct statement. 90 out of the 100 intervals will contain the mean. O It is a correct statement. This is guaranteed by the definition of confidence interval. It is not a correct statement. We expect 95 out of 100 intervals will contain the mean, but we don't know this to be true. O It is a correct statement. Since we are taking the same sample, we expect all intervals to contain the mean.

Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let μ denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (7.6, 9.1). (a) Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning. O The 90% would be the same since the z critical value for 90% is the same as the z critical value for 95%. The 90% would be narrower since the z critical value for 90% is larger than the z critical value for 95%. The 90% would be wider since the z critical value for 90% is larger than the z critical value for 95%. O The 90% would be narrower since the z critical value for 90% is smaller than the z critical value for 95%. O The 90% would be wider since the z critical value for 90% is smaller than the z critical value for 95%. (b) Consider the following statement: There is a 95% chance that is between 7.6 and 9.1. Is this statement correct? Why or why not? O It is not a correct statement. We are 95% confident in the general procedure for creating the interval, but the mean may or may not be enclosed in this interval. O It is not a correct statement. Each interval contains the mean by definition. O It is a correct statement. Each interval contains the mean by definition. O It is a correct statement. There is only a 5% chance that the mean is not between these values. It is not a correct statement. There is a 5% chance that the mean is between these values. (c) Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.6 and 9.1. Is this statement correct? Why or why not? O It is not a correct statement. The interval is an estimate for the sample mean, not a boundary for sample values. It is not a correct statement. The interval is an estimate for the sample mean, not a boundary for population values. O It is a correct statement. This interval is a great estimate of boundaries for population values. O It is a correct statement. This is the definition of a confidence interval. It is not a correct statement. The interval is an estimate for the population mean, not a boundary for population values. (d) Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% interval is repeated 100 times, 95 of the resulting intervals will include u. Is this statement correct? Why or why not? O It is not a correct statement. We expect 5 out of the 100 intervals to contain the mean. O It is not a correct statement. 90 out of the 100 intervals will contain the mean. O It is a correct statement. This is guaranteed by the definition of confidence interval. It is not a correct statement. We expect 95 out of 100 intervals will contain the mean, but we don't know this to be true. O It is a correct statement. Since we are taking the same sample, we expect all intervals to contain the mean.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Please solve the a, b, c, and d.

Transcribed Image Text:Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let u denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval is (7.6, 9.1). (a) Would a 90% confidence interval calculated from this same sample have been narrower or wider than the given interval? Explain your reasoning. The 90% would be the same since the z critical value for 90% is the same as the z critical value for 95%. The 90% would be narrower since the z critical value for 90% is larger than the z critical value for 95%. The 90% would be wider since the z critical value for 90% is larger than the z critical value for 95%. The 90% would be narrower since the z critical value for 90% is smaller than the z critical value for 95%. The 90% would be wider since the z critical value for 90% is smaller than the z critical value for 95%. (b) Consider the following statement: There is a 95% chance that u is between 7.6 and 9.1. Is this statement correct? Why or why not? It is not a correct statement. We are 95% confident in the general procedure for creating the interval, but the mean may or may not be enclosed in this interval. It is not a correct statement. Each interval contains the mean by definition. It is a correct statement. Each interval contains the mean by definition. It is a correct statement. There is only a 5% chance that the mean is not between these values. It is not a correct statement. There is a 5% chance that the mean is between these values. (c) Consider the following statement: We can be highly confident that 95% of all bottles of this type of cough syrup have an alcohol content that is between 7.6 and 9.1. Is this statement correct? Why or why not? It is not a correct statement. The interval is an estimate for the sample mean, not a boundary for sample values. It is not a correct statement. The interval is an estimate for the sample mean, not a boundary for population values. It is a correct statement. This interval is a great estimate of boundaries for population values. It is a correct statement. This is the definition of a confidence interval. It is not a correct statement. The interval is an estimate for the population mean, not a boundary for population values. (d) Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% interval is repeated 100 times, 95 of the resulting intervals will include u. Is this statement correct? Why or why not? It is not a correct statement. We expect 5 out of the 100 intervals to contain the mean. It is not a correct statement. 90 out of the 100 intervals will contain the mean. It is a correct statement. This is guaranteed by the definition of confidence interval. It is not a correct statement. We expect 95 out of 100 intervals will contain the mean, but we don't know this to be true. It is a correct statement. Since we are taking the same sample, we expect all intervals to contain the mean.

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