DEPARTMENT OF ECONOMICS AND RELATED STUDIESStatistics and Econometrics (ECO00037)

Tutorial Exercise 3

Term 1: Topics covered in Lecture 5

1. Suppose that random variable X has mean and variance 2. Random samples,X1,X2,...,Xn, will be drawn from the population by a researcher who does notknow the values of and 2. The researcher considers two estimators of , denedby

^n =

Pni=1Xin

and ~n =8 +

Pni=1Xin

.

(a) Obtain E(^n) and E(~n) in terms of and n. Are they unbiased estimators?

(b) Obtain var(^n) and var(~n) in terms of 2 and n.

(c) Suppose you know (but the researcher does not) 2 = 4. Obtain mean squareerrors of ^10 (i.e. n = 10) and ~20 (i.e. n = 20). Comment on these twoestimators.

(d) Are ^n and ~n consistent?

2. A simple random sample of 15 pupils attending a certain school is found to havean average IQ of 107.3 with a sample variance of 32.5.

(a) Calculate a 95% condence interval for the unknown population mean IQ,stating any assumptions you need to make. Interpret this interval.

(b) Explain whether you would be happy with a parents claim that the averageIQ at the school is 113.

3. You have been asked to measure the respiratory health of patients attending aGPs surgery both on 1st of October and 1st of December of a particular year. Thepatients respiratory health is measured by the forced expiratory volume (FEV) inone second. The experiment involves measuring the FEV of 45 patients selectedat random from the appointment list, which gives a sample mean and a samplestandard deviation of the dierences of ith patient, di = fevdecember;i fevoctober;i,d, = 0:13, sd = 1:5.

(a) Stating any assumptions you need to make, nd the 95% condence intervalfor the population mean dierence between those two days.

(b) Explain whether you would be happy with a claim that there is no dierencein the population means of FEV of a patient, between these two days.

1

4. In an opinion poll based on 100 interviews in each city A and B, 34 people in thecity A and 26 people in the city B say they are not satised with the level of theirlocal Council services.

(a) Stating any assumptions you need to make, nd a 95% condence interval forthe true proportion of people in each city A and B separately, who are notsatised with local council services.

(b) Explain whether you would be happy with a claim that the true proportion ofthe unsatised people with local council services in the city A is higher thatin the city B. (use 95% condence interval)

2