Suppose we are interested in studying a population to estimate its mean. The population is normal and has a standard deviation of o=19. We have taken a random sample of size n=87 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 = 106.5. Also shown are the lower and upper limits of the 80% 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 u=105, 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=87 from this same population. (The 80% 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. 80% 80% 95% 95% lower upper lower upper limit limit limit limit 109.1 110.5 S1 106.5 103.9 102.5 S2 103.6 101.0 106.2 99.6 107.6 S3 104.1 101.5 106.7 100.1 108.1 S4 107.3 104.7 109.9 103.3 111.3 S5 105.5 102.9 108.1 101.5 109.5 S6 108.3 105.7 110.9 104.3 112.3 S7 104.5 101.9 107.1 100.5 108.5 S8 108.5 105.9 111.1 104.5 112.5 S9 103.7 101.1 106.3 99.7 107.7 |S10 109.8| 107.2 | 112.4 | 105.8 | 113.8 |S11 106.4|103.8 | 109.0 | 102.4 | 110.4 S12 106.4 103.8 109.0 102.4 110.4 S13 101.5 98.9 104.1 97.5 105.5 S14 105.6 103.0 108.2 101.6 109.6 |S15 104.6| 102.0 | 107.2 | 100.6 | 108.6 S16 104.5 101.9 107.1 100.5 108.5 S17 101.7 99.1 104.3 97.7 105.7 S18 100.4 97.8 103.0 96.4 104.4 S19 106.1 103.5 108.7 102.1 110.1 S20 103.4 100.8 106.0 99.4 107.4 x 80% confidence intervals |+++ 96.0 114.0 96.0 95% confidence intervals ||||| 114.0 (a) How many of the 80% confidence intervals constructed from the 20 samples contain the population mean, u = 105? (b) How many of the 95% confidence intervals constructed from the 20 samples contain the population mean, μ=105? (c) Choose ALL that are true. The center of the 80% confidence interval for Sample 1 is 106.5, because the center of a confidence interval for the population mean must be the sample mean. Since Sample 19 and Sample 20 are drawn from the same population, the center of the 95% confidence interval for Sample 19 must be the same as the center of the 95% confidence interval for Sample 20. For each sample, the 80% confidence interval for the sample is included in the 95% confidence interval for the sample. 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. None of the choices above are true.
Suppose we are interested in studying a population to estimate its mean. The population is normal and has a standard deviation of o=19. We have taken a random sample of size n=87 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 = 106.5. Also shown are the lower and upper limits of the 80% 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 u=105, 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=87 from this same population. (The 80% 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. 80% 80% 95% 95% lower upper lower upper limit limit limit limit 109.1 110.5 S1 106.5 103.9 102.5 S2 103.6 101.0 106.2 99.6 107.6 S3 104.1 101.5 106.7 100.1 108.1 S4 107.3 104.7 109.9 103.3 111.3 S5 105.5 102.9 108.1 101.5 109.5 S6 108.3 105.7 110.9 104.3 112.3 S7 104.5 101.9 107.1 100.5 108.5 S8 108.5 105.9 111.1 104.5 112.5 S9 103.7 101.1 106.3 99.7 107.7 |S10 109.8| 107.2 | 112.4 | 105.8 | 113.8 |S11 106.4|103.8 | 109.0 | 102.4 | 110.4 S12 106.4 103.8 109.0 102.4 110.4 S13 101.5 98.9 104.1 97.5 105.5 S14 105.6 103.0 108.2 101.6 109.6 |S15 104.6| 102.0 | 107.2 | 100.6 | 108.6 S16 104.5 101.9 107.1 100.5 108.5 S17 101.7 99.1 104.3 97.7 105.7 S18 100.4 97.8 103.0 96.4 104.4 S19 106.1 103.5 108.7 102.1 110.1 S20 103.4 100.8 106.0 99.4 107.4 x 80% confidence intervals |+++ 96.0 114.0 96.0 95% confidence intervals ||||| 114.0 (a) How many of the 80% confidence intervals constructed from the 20 samples contain the population mean, u = 105? (b) How many of the 95% confidence intervals constructed from the 20 samples contain the population mean, μ=105? (c) Choose ALL that are true. The center of the 80% confidence interval for Sample 1 is 106.5, because the center of a confidence interval for the population mean must be the sample mean. Since Sample 19 and Sample 20 are drawn from the same population, the center of the 95% confidence interval for Sample 19 must be the same as the center of the 95% confidence interval for Sample 20. For each sample, the 80% confidence interval for the sample is included in the 95% confidence interval for the sample. 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. None of the choices above are true.
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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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 a=19. We have taken a
random sample of size n=87 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 = 106.5. Also shown are the lower and upper limits of the 80% 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 = 105, 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=87 from this same population. (The 80% 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.
80% 80% 95% 95%
lower upper lower upper
limit limit limit limit
110.5
S1 106.5 103.9 109.1 102.5
S2 103.6 101.0 106.2 99.6 107.6
S3 104.1 101.5 106.7 100.1 108.1
S4 107.3 104.7 109.9 103.3 111.3
S5 105.5 102.9 108.1 101.5 109.5
S6 108.3 105.7 110.9 104.3 112.3
S7 104.5 101.9 107.1 100.5 108.5
S8 108.5 105.9 111.1 104.5 112.5
S9 103.7 101.1 106.3 99.7 107.7
S10 109.8| 107.2 | 112.4 105.8 113.8
S11 106.4| 103.8 | 109.0 | 102.4 | 110.4
S12 106.4 103.8 109.0 102.4 110.4
S13 101.5 98.9 104.1 97.5 105.5
S14 105.6 103.0
108.2 101.6 109.6
S15 104.6| 102.0 107.2 100.6 108.6
S16 104.5 101.9 107.1 100.5 108.5
S17 101.7 99.1 104.3
97.7
105.7
S18 100.4 97.8 103.0 96.4
104.4
S19 106.1 103.5 108.7 102.1 110.1
S20 103.4 100.8 106.0 99.4 107.4
80% confidence intervals
HHH
96.0
++
H
114.0 96.0
95% confidence intervals
|||||||
114.0
(a) How many of the 80% confidence intervals constructed from the 20 samples contain the population mean, u = 105?
(b) How many of the 95% confidence intervals constructed from the 20 samples contain the population mean, μ = 105?
(c) Choose ALL that are true.
The center of the 80% confidence interval for Sample 1 is 106.5, because the center of a confidence interval for the
population mean must be the sample mean.
Since Sample 19 and Sample 20 are drawn from the same population, the center of the 95% confidence interval for
Sample 19 must be the same as the center of the 95% confidence interval for Sample 20.
For each sample, the 80% confidence interval for the sample is included in the 95% confidence interval for the
sample.
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.
None of the choices above are true.
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