STA 9700: Homework 2
Read STA 9700 Lecture Notes 2; Read again, write questions in margins. (There is some related material in Kutner, pg. 2-27.) STA 9708 LN 5 (Expectation and variance of random variables)
Questions based on STA 9700 Lecture Notes 2
2.1 Looking at Fig. 2.1 in Lecture Notes 2, we see that there is a general rise in the NetWt of the bags as the Count increases. While the phrase "general rise" is not clearly defined, it is certainly better than the following commonplace description, "Bags with more M&M's are heavier." That statement is far too simplistic!
(a) The data for Fig. 2.1 is shown on pages 15-17 of Lecture Notes 2. Using the data, give several examples of pairs of bags for which the statement "Bags with more M&M's are heavier" is false.
(b) Having shown that not all bags with more M&M's are heavier than all bags with fewer M&M's, consider this next vague description, "The average bag containing 18 M&M's weighs more than the average bag containing 17 M&M's." What is vague about that statement? Hint: which bag is the average bag? What is the definition of the average bag? (That is as hard as defining or locating the average American, which should be easy because we hear about that dude everyday on the news.)
(c) Critique this statement: “Since on page 12 the sample slope is 1.276 when regressing net weight on count for the 192 bags, then the sample average for bags with Count=18 must be higher than for Count=17.” And, find a counterexample in the data set, itself!
(d) What statement are we struggling to make here about the relationship between the sub-populations of Net Weights and their Count?
2.2 Putting together the BigMM SAS program and the following Proc Reg routine, we can create a SAS program that computes the sample slope, the sample intercept, and the root mean square error for each of the 8 groups of bags of M&M's (there are 24 bags per group), outputs those statistic to a SAS file, and prints the file.
proc reg outest=LTatum; model NetWt=Count; By Group; run; proc print data=LTatum; run;
The Proc Reg option "outest=LTatum" instructs SAS to save the regression statistics (or "estimates") into a SAS file named "Ltatum." The output is shown below due to difficulties with SAS, but I would be delighted if you are able to produce it yourself! The sample slopes are in the Count column.
Obs Group _MODEL_ _TYPE_ _DEPVAR_ _RMSE_ Intercept Count Wt 1 2 MODEL1 PARMS NetWt 1.52202 25.2154 1.28176 -1 2 3 MODEL1 PARMS NetWt 0.94023 27.2769 1.16531 -1 3 5 MODEL1 PARMS NetWt 0.96081 17.6571 1.65238 -1 4 6 MODEL1 PARMS NetWt 1.01435 19.1121 1.59741 -1 5 7 MODEL1 PARMS NetWt 1.53226 26.1459 1.22875 -1 6 8 MODEL1 PARMS NetWt 1.09972 28.7744 1.11778 -1 7 9 MODEL1 PARMS NetWt 0.99709 22.1760 1.42708 -1 8 10 MODEL1 PARMS NetWt 1.10568 26.5912 1.18456 -1
(a) You now have 8 different sample slopes, or 8 different values for . These can be viewed as 8 values drawn from what population? (Hint: You need The Story of Many Possible Samples.)
(b) Imagine that for our production run of 10,000 bags of Peanut M&M's that we regressed the 10,000 net weights on their respective 10,000 counts. What would we call the resulting intercept and slope? Show the answer in words and Greek letters.
(c) Using The Story of Many Possible Samples, explain what it would mean to say that is an unbiased estimator.
2.3 Refer to the SAS output on page 12, for the regression using all 192 bags.
(a) Compute the value for count=18.
(b) What is estimated by b1?
(c) How is the value related to ?
Expected Value and Variance Review Questions
2.4 For a roll of a fair die with 4 sides, numbered 1 to 4, find the expected value and the variance.
2.5 Find the probability distribtuion for the average of two rolls of a fair die with four sides. Then, compute expected value and variance of the average from the distribution.
2.6 How were the answers to question 2.5 related to those of question 2.4?
2.7 Generic Calculus Questions; warming up to least squares: Find the derivative with respect to x of the following functions:
(a) y = x2 (b) y = (4x + 3)2 (c) y = (-3x2 + x)
2.8 The R function
will regress y on x, and the function
produces output similar to the SAS regression output. For BigMM, see if you can get output with similar values as those given by SAS on page 16. Locate the estimate of the variance of epsilon.