UCSB PSTAT 109 Statistics for Economics Question-Quiz 7

Problem 1

 For the following data,

21, 18, 17, 23, 22, 19, 21, 17, 25,

compute the following measures:

a. mean,

b. sample variance,

c. the 30th percedntile,

d. the median,

e. the 83rd percentile,

In general, what is the advantage of reporting the median value as opposed to the mean

value?

Problem 2

a. Suppose X¯1 and S12 are the sample mean and sample variance from a population with

mean µ1 and variance σ 12 , and X¯2 and S22 are the sample mean and sample variance

from a second independent population with mean µ2 and σ22. The sample sizes are n1

and n2 , respectively. Show that X¯1 − X¯2 is an unbiased estimator of µ1 − µ2 .

b. Now suppose we have the same setup as in (a) but σ12 = σ22 = σ 2 . Show that the

pooled sample variance

is an unbiased estimator of σ 2 . (Hint: Sample variance S 2 is an unbiased estimator of

population variance σ 2.)

Problem 3

 Candidate A is running for student body president at some university. The

proportion of students who favor this candidate at the moment is 0.8.

a. Suppose a simple random sample of 100 students is taken. What are the expected?

value, standard deviation, and shape of the sampling distribution of p¯?

b. If a simple random sample of 100 students is taken, what is the probability that the

number of students in the sample who DO NOT favor Candidate A will be between

26 and 30?