Stat 11 Spring 2011 - Homework 4
(1) A general can plan a campaign to fight one major battle or three small battles. He believes that he has probability .56 of winning the large battle and probability .78 of winning each of the small battles. Assume that victories or defeats in the small battles are independent. The general must win either the large battle or all three small battles to win the campaign. Which strategy should he choose?
(2) Suppose a drug test is 97% accurate. All 150 employees at a company are made to take the drug test. If none of the 150 employees are in fact using drugs, what is the probability that at least one employee will be wrongly accused as a result of drug testing?
(3-8) The following questions deal with the computer simulation from the class of Wednesday, 2/16. Our goal was to estimate the number of people in a large class without having to count each person individually.
Notation: Let N represent the number of people in the class. Let n represent the number of people who were randomly sampled.
We compared three methods of estimating N. Depending on which section you're in, these may have included the following methods (the numbers may not match what we called them in class):
Method 1: Take the average of the n numbers in the sample and double it
Method 2: Take the average of the n numbers and add 2 SD's
Method 3: Multiply the largest of the n numbers in the sample by n+1/n
We used the computer to take samples of size n = 5 from a hypothetical large class with size N = 100, and recorded the estimates from each method. The sampling distributions are shown below:
(3) In this specific context of estimating N, explain what a sampling distribution is. Why is it important to look at the sampling distributions in comparing the three methods?
(4) I claimed in class that method 3 is the "best". Based on the sampling distribution histograms, describe two criteria that the "best" method should satisfy.
(5) What should happen to the SD of the sampling distributions as n increases (assuming N does not change)? (Note: be sure you address the SD of the sampling distributions, not the SD of the values in the sample.) Explain briefly.
(6) Try applying Method 1 to the sample {44, 92, 12, 21, 56}. What is your prediction for N? Why is this prediction not sensible?
(7) The drawback in Method 1 pointed out in question (12) can be fixed as follows: Method 1a: Take the average of the n values in the sample and double it. This is our estimate of N. However, if this estimate is smaller than the largest value in the sample, then use that largest value as our estimate of N instead.
Using the sample given in (12) with Method 1a, how many people would you estimate are in the class?
(8) We saw that Method 1 is unbiased. Is Method 1a unbiased? Explain why or why not.