## Statistics 101 Problem Set #7 Fall 1999

This homework is to be done individually. It is due on the last day of class.
1. Exercise 7.4 page 268
2. Exercise 7.5 page 268

1. Taxicabs are numbered from 1,2,...,N. We want to estimate N, the total number of taxicabs. We will observe n (n less than N) cabs. Denote their number by Y1,...,Yn. We consider two estimators for N:

1. Estimator 1 is (2 Y-bar - 1) where Y-bar is the average of all the Y's observed.
2. Estimator 2 is Maximum of Y1,...,Yn.

Assume that the n taxicabs that we will see is a simple random sample of the N cabs. (Sampling without replacement). We want to compare estimator 1 to estimator 2. To this end, assume that N = 4 and n = 2.
1. Write down the 4 choose 2 possible samples of taxi numbers.
2. Write down the sampling distribution of teach of the estimators. (Sketch the probability distribution function.)
3. Which of the estimators is unbiased? Note that the true answer is N = 4.
4. Which estimator is closer to 4 on the average, where closeness is measured by

(estimate - 4)2?
1. Exercise 7.15 page 273
2. Exercise 7.16 page 273
3. Exercise 7.17 page 274
1. Exercise 7.20 page 276
2. Exercise 7.21 page 276
1. Exercise 7.25 page 279
2. Exercise 7.26 page 279
3. Exercise 7.27 page 279
1. Exercise 7.72 page 304
2. Exercise 7.74 page 304