Assignment for January 25, 2006

Assignment for Wednesday, January 25

Please Note: This assignment includes problems to turn in. (See Part III below.)

Reminder: Students whose last names begin with A through L will be called on Wednesday.

I. a. Read Section 1.3 (pp. 64 - 82), paying attention to the usual (cautions, definitions, etc.) and the following:

Much of this section will be review of material from Probability and from the reading from Friday on random variables. But be sure not to let this lull you into thinking you understand it all; there are some differences in the probability and statistics approaches that are important -- for example, the first paragraph.
If you find the description of normal quantile plots onp. 80 confusing, you're not the only one. But be sure to go on to pp. 81 - 83 to understand how these plots are used.

b. Also read the handout Normal Quantile Plots for more explanation of these plots.

II. Do the following exercises to reinforce the reading and for possible class discussion: 1.79, 1.80, 1.83, 1.87, 1.88, 1.90, 1.99, 1.104, 1.116, 1.117, 1.119, 1.121, 1.23. (See Using Minitab in M 358K for how to make a normal quantle plot for Exercise 1.23 in Minitab.)

III. Turn in: -> Be sure to review the Guidelines for Written Homework on the First Day Handout before you write up your problems to hand in.

1. (Review of some probability concepts) You may use the following facts from probability in this problem:

E(X + Y) = E(X) + E(Y), where X and Y are random variables and E(X) denotes the expected value (=expectation = mean) of the random variable X.

If c is a constant and X is a random variable, then E(cX) = cE(X).

If X is a constant random variable (with constant value c), then E(X) = c.

The variance of the random variable X is defined to be Var(X) = E([X - E(X)]²)

    a. You have n random variables X₁, X₂, ..., X_n that all have the same distribution with mean µ. Define the new random variable X-bar to be (1/n)( X₁+ X₂,+ ...+ X_n). (In other words, X-bar is the "average" of X₁, X₂, ..., X_n). Find the mean E(X-bar) of X-bar, and use some of the facts above to prove that your answer is correct.
    b. Use the definition of variance above and the facts about expected value above to derive a formula for Var( c + X), where c is a constant. (Be sure to include all steps in your derviation of the formula.)
    c. Use the definition of variance above and the facts about expected value above to derive a formula for Var( cX), where c is a constant. (Be sure to include all steps in your derviation of the formula.)
    d. Use the definition of variance above and the facts about expected value above to derive the following alternate formula for variance: Var(X) = E(X²) - [E(X)]². (Be sure to include all steps in your derviation of the formula.)

2. a. Exercise 1.30 on p. 35. You will need to use computer software for this. Be sure to hand in all three of the histograms requested. If you are using Minitab, you can find instructions in the handout Using Minitab in M 358K. The data are in the file ex01_030.xxx. If you are using Minitab in the math department computer lab, see the handout Using Minitab in M 358K for how to access this file in the math department computer system. If you are working elsewhere, you will need to get the file from the CD that comes with your book, or download the files from the book web site.
    b. Also make (and hand in) a stemplot of these data. You may do this either by hand or by computer.