(Central Limit Theorem.) Consider the experiment of tossing a fair die once. Let X denote the number that occurs. Theoretically, X follows a discrete uniform distribution.

(a) Find the theoretical density, mean, and variance for X.

(b) Now consider an experiment in which the die is tossed 20 times and the results averaged. By the Central Limit Theorem, what is the theoretical mean and variance for the random variable X?

(c) Perform the experiment of part (b) 25 times and record the value of X each time. (You will toss the die 500 times and obtain a data set that consists of 25 averages.) What shape should the stem-and-leaf diagram for these data assume? Explain. Construct a stem-and-leaf diagram for your data. Did the diagram take the shape that you expected?

(d) Approximately what value would you expect to obtain if you averaged the data of part (c)? Average your 25 observations on Y. Did the result come out as expected?

(e) Approximately what value would you expect to obtain if you found the sample variance for the data of part (c)? Explain. Find s2 for your 25 observations on X. Did the result come out as expected?

(f) If you were to construct 95% confidence intervals on tt based on each of the values of X found in part (c), approximately how many of them would you expect to contain the true value of i.t? From your data, can you find an example of a confidence interval that does contain A? of a confidence interval that does not contain µ