Applet Exercise Refer to Exercise 7.1. Use the applet

Chapter 7, Problem 3E

(choose chapter or problem)

Applet Exercise Refer to Exercise 7.1. Use the applet DiceSample and scroll down to the next part of the screen that corresponds to taking samples of size n = 12 from the population corresponding to tossing a balanced die.

Take a single sample of size n = 12 by clicking the button “Roll One Set.” Use the button “Roll One Set” to generate nine more values of the sample mean. How does the histogram of observed values of the sample mean compare to the histogram observed in Exercise 7.1(c) that was based on ten samples each of size 3?Use the button “Roll 10 Sets” nine more times until you have obtained and plotted 100 realized values (each based on a sample of size n = 12) for the sample mean \(\bar{Y}\) Click on the button “Show Stats” to see the mean and standard deviation of the 100 values

\(\left(\bar{y}_{1}, \bar{y}_{2}, \ldots, \bar{y}_{100}\right)\) that you observed.

i.         How does the average of these 100 values of \(\bar{y}_{i}, i=1,2, \ldots, 100\) compare to the

                 average of the 100 values (based on samples of size n = 3) that you obtained in

Exercise 7.1(d)?  

Ii.         Divide the standard deviation of the 100 values of \(\bar{y}_{i}, i=1,2, \ldots, 100\) based on

samples of size 12 that you just obtained by the standard deviation of the 100   values (based on samples of size n = 3) that you obtained in Exercise 7.1. Why do you expect to get a value close to 1/2? [Hint: \(\mathrm{V}(\bar{Y})=\sigma^{2} / \mathrm{n}\).]

Click on the button “Toggle Normal.” The (green) continuous density function plotted over the histogram is that of a normal random variable with mean and standard deviation equal to the mean and standard deviation of the 100 values, \(\left(\bar{y}_{1}, \bar{y}_{2}, \ldots, \bar{y}_{100}\right)\), plotted on the histogram. Does this normal distribution appear to reasonably approximate the distribution described by the histogram?

Equation Transcription:

Text Transcription:

\bar Y

( \bar y_1, \bar y_2,...,,\bar y_100)

\bar y_i, i=1,2,...100

\bar y_i, i=1,2,...100

V(\bar Y)\sigma^ 2/n

( \bar y_1, \bar y_2,...,,\bar y_100)