Solved: Applet Exercise What does the sampling
Chapter 7, Problem 5E(choose chapter or problem)
Applet Exercise What does the sampling distribution of the sample mean look like if samples are taken from an approximately normal distribution? Use the applet Sampling Distribution of the Mean (at www.thomsonedu.com/statistics/wackerly) to complete the following. The population to be sampled is approximately normally distributed with \(\mu=16.50 \text { and } \sigma=6.03\) (these values are given above the population histogram and denoted M and S, respectively).
Use the button “Next Obs” to select a single value from the approximately normal popu- lation. Click the button four more times to complete a sample of size 5. What value did you obtain for the mean of this sample? Locate this value on the bottom histogram (the histogram for the values of \(\bar{Y}\)Click the button “Reset” to clear the middle graph. Click the button “Next Obs” five more
times to obtain another sample of size 5 from the population. What value did you obtain for the mean of this new sample? Is the value that you obtained equal to the value you obtained in part (a)? Why or why not?
Use the button “1 Sample” eight more times to obtain a total of ten values of the sample mean. Look at the histogram of these ten means.What do you observe?How does the mean of these 10 \(\bar{y}\) -values compare to the population mean \(\mu\)?Use the button “1 Sample” until you have obtained and plotted 25 realized values for the sample mean \(\bar{Y}\) , each based on a sample of size 5.What do you observe about the shape of the histogram of the 25 values of\(\bar{y}_{i}, \mathrm{i}=1,2, \ldots, 25\)?
How does the value of the standard deviation of the 25 \(\bar{y}\) values compare with the theoretical value for\(\mathrm{V}(\bar{Y})=\sigma^{2} / \mathrm{n}\)?
Click the button “1000 Samples” a few times, observing changes to the histogram as you generate more and more realized values of the sample mean. What do you observe about the shape of the resulting histogram for the simulated sampling distribution of \(\bar{Y}\)?Click the button “Toggle Normal” to overlay (in green) the normal distribution with the same mean and standard deviation as the set of values of \(\bar{Y}\) that you previously generated. Does this normal distribution appear to be a good approximation to the sampling distribution of \(\bar{Y}\) ?Equation Transcription:
Text Transcription:
\mu =16.50 and \sigma =6.03
\bar Y
\bar y
\mu
\bar Y
\bar y_i, i=1,2,...25
\bar y
\sigma^y
\bar Y
V(\bar Y)=\sigma^2/n
\bar Y
\bar Y
\bar Y
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