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Suppose we are interested in studying a population to estimate its mean. The population is normal and has a standard deviation of o=9. We have taken a random sample of size n-60 from the population. This is Sample 1 in the table below. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.) As shown in the table, the sample mean of Sample 1 is x = 81.1. Also shown are the lower and upper limits of the 75% confidence interval for the population mean using this sample, as well as the lower and upper limits of the 95% confidence interval. Suppose that the true mean of the population is μ = 80, which is shown on the displays for the confidence intervals. Press the "Generate Samples" button to simulate taking 19 more random samples of size n=60 from this same population. (The 75% and 95% confidence intervals for all of the samples are shown in the table and graphed.) Then complete parts (a) through (c) below the table.

Suppose we are interested in studying a population to estimate its mean. The population is normal and has a standard deviation of o=9. We have taken a random sample of size n-60 from the population. This is Sample 1 in the table below. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.) As shown in the table, the sample mean of Sample 1 is x = 81.1. Also shown are the lower and upper limits of the 75% confidence interval for the population mean using this sample, as well as the lower and upper limits of the 95% confidence interval. Suppose that the true mean of the population is μ = 80, which is shown on the displays for the confidence intervals. Press the "Generate Samples" button to simulate taking 19 more random samples of size n=60 from this same population. (The 75% and 95% confidence intervals for all of the samples are shown in the table and graphed.) Then complete parts (a) through (c) below the table.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Suppose we are interested in studying a population to estimate its mean. The population is normal and has a standard deviation of \(\sigma = 9\). We have taken a random sample of size \(n = 60\) from the population. This is Sample 1 in the table below. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.) As shown in the table, the sample mean of Sample 1 is \(\bar{x} = 81.1\). Also shown are the lower and upper limits of the 75% confidence interval for the population mean using this sample, as well as the lower and upper limits of the 95% confidence interval. Suppose that the true mean of the population is \(\mu = 80\), which is shown on the displays for the confidence intervals. Press the "Generate Samples" button to simulate taking 19 more random samples of size \(n = 60\) from this same population. (The 75% and 95% confidence intervals for all of the samples are shown in the table and graphed.) Then complete parts (a) through (c) below the table. ### Table of Sample Means and Confidence Intervals | | \(\bar{x}\) | 75% lower limit | 75% upper limit | 95% lower limit | 95% upper limit | |-------|-------------|----------------|----------------|----------------|----------------| | S1 | 81.1 | 79.8 | 82.4 | 78.8 | 83.4 | | S2 | 80.8 | 79.5 | 82.1 | 78.5 | 83.1 | | S3 | 80.1 | 78.8 | 81.4 | 77.8 | 82.4 | | S4 | 79.3 | 78.0 | 80.6 | 77.0 | 81.6 | | S5 | 81.0 | 79.7 | 82.3 | 78.7 | 83.3 | | S6 | 81.8 | 80.5 | 83.1
Transcribed Image Text:Suppose we are interested in studying a population to estimate its mean. The population is normal and has a standard deviation of \(\sigma = 9\). We have taken a random sample of size \(n = 60\) from the population. This is Sample 1 in the table below. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.) As shown in the table, the sample mean of Sample 1 is \(\bar{x} = 81.1\). Also shown are the lower and upper limits of the 75% confidence interval for the population mean using this sample, as well as the lower and upper limits of the 95% confidence interval. Suppose that the true mean of the population is \(\mu = 80\), which is shown on the displays for the confidence intervals. Press the "Generate Samples" button to simulate taking 19 more random samples of size \(n = 60\) from this same population. (The 75% and 95% confidence intervals for all of the samples are shown in the table and graphed.) Then complete parts (a) through (c) below the table. ### Table of Sample Means and Confidence Intervals | | \(\bar{x}\) | 75% lower limit | 75% upper limit | 95% lower limit | 95% upper limit | |-------|-------------|----------------|----------------|----------------|----------------| | S1 | 81.1 | 79.8 | 82.4 | 78.8 | 83.4 | | S2 | 80.8 | 79.5 | 82.1 | 78.5 | 83.1 | | S3 | 80.1 | 78.8 | 81.4 | 77.8 | 82.4 | | S4 | 79.3 | 78.0 | 80.6 | 77.0 | 81.6 | | S5 | 81.0 | 79.7 | 82.3 | 78.7 | 83.3 | | S6 | 81.8 | 80.5 | 83.1
(a) How many of the 75% confidence intervals constructed from the 20 samples contain the population mean, μ = 80? □ (b) How many of the 95% confidence intervals constructed from the 20 samples contain the population mean, μ = 80? □ (c) Choose ALL that are true. - □ It is not surprising that some 75% confidence intervals are different from other 75% confidence intervals. Each confidence interval depends on its sample, and different samples may give different confidence intervals. - □ All of the 95% confidence intervals should be the same as each other. Since they are not all the same, there must have been errors due to rounding. - □ The center of the 75% confidence interval for Sample 1 is 80, because the center of any confidence interval for the population mean must be the population mean. - □ We would expect to find more 75% confidence intervals that contain the population mean than 95% confidence intervals that contain the population mean. Given a sample, a higher confidence level results in a narrower interval. - □ None of the choices above are true.
Transcribed Image Text:(a) How many of the 75% confidence intervals constructed from the 20 samples contain the population mean, μ = 80? □ (b) How many of the 95% confidence intervals constructed from the 20 samples contain the population mean, μ = 80? □ (c) Choose ALL that are true. - □ It is not surprising that some 75% confidence intervals are different from other 75% confidence intervals. Each confidence interval depends on its sample, and different samples may give different confidence intervals. - □ All of the 95% confidence intervals should be the same as each other. Since they are not all the same, there must have been errors due to rounding. - □ The center of the 75% confidence interval for Sample 1 is 80, because the center of any confidence interval for the population mean must be the population mean. - □ We would expect to find more 75% confidence intervals that contain the population mean than 95% confidence intervals that contain the population mean. Given a sample, a higher confidence level results in a narrower interval. - □ None of the choices above are true.
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