Econometrics

Problem 1

Supopse we ask students in Econ 423 how many siblings they have and how far away from

Corvallis they grew up.

(a) Assume the population of interest is all students at OSU, and the sample means from the

survey questions are used to estimate the respective population means. Discuss whether non-

sampling error is a major problem with the estimates.

(b) Now assume the population of interest is all American adults. Discuss whether non-sampling

error is likely a major problem with the estimates.

(c) Explain why the presence of non-sampling error depends on whether the population of

interest is all OSU students or all American adults.

Problem 2

Suppose you have a random sample of size n on a random variable X, and you are interested

in estimating the population mean, µx. Consider two different estimators: (i) the sample mean,

X=1 n

n∑ i=1

Xi, and (ii) another estimator, X̃ = 1 2 X1 +

1 2(n−1)

n∑ i=2

Xi.

(a) Show that both X and X̃ are unbiased estimators of the population mean.

(b) Calculate Var(X) and Var(X̃ ). Which one has the lower variance?

Hint: You will use the fact that 1 n < 1

4

( n n−1

) for n>2.

(c) Show that X is consistent, and show that X̃ is not consistent.

(d) Explain why it is incorrect to say, “X is more effi cient than X̃ .”

(e) Give an example of an estimator that has a lower variance than X.

1

 

 

Problem 3

Suppose you have a random sample of size n= 50 for a random variable, X, with observed x =2.4, and you know that σ2x=2.

(a) Construct an approximate 95% Confidence Interval estimate of the population mean, µx.

(b) Construct an approximate 99% Confidence Interval estimate of the population mean, µx.

(c) Explain (in words – do not actually do it) how you would solve (a) and (b) differently using

s2x=2 instead of σ 2 x. In other words, you only have an estimate of the variance of X. Would you

expect the confidence intervals to be wider or shorter?

 

Problem 4

Set up the rejection region, compute the p-value, and interpret the result of the following tests

(a) H0 : µx = 1000, H1 : µx 6= 1000, σx = 200, n = 100, x = 980, α = 0.01 (b) H0 : µx = 50, H1 : µx > 50, σx = 5, n = 9, x = 51, α = 0.01

(c) H0 : µx = 15, H1 : µx < 15, σx = 2, n = 25, x = 14.3, α = 0.10

(d) H0 : µx = 50, H1 : µx < 50, σx = 15, n = 100, x = 52, α = 0.05

 

Problem 5

A marketing firm hires a researcher to investigate mobile-phone usage patterns among OSU students. She randomly samples 100 students, 40 male and 60 female. The male students in

the sample averaged 90 minutes usage per week, with a variance of 400 (minutes per week)2. The

females averaged 200 minutes usage per week with a variance of 1200 (minutes per week)2. Assume

the variances are known (so you can use a normal distribution approximation).

(a) Test the hypothesis that female students average 100 more minutes usage per week than the

male students, against the alternative that this difference is more than 100 minutes. Use an 5%

test size.

(b) Estimate an approximated 95% confidence interval of the difference in the average number

of minutes per week used by females and males.

(c) Comment on the following statement: “Since the inferential exercise is comparing average

mobile-phone usage between females and males, a better sampling strategy would have been to

sample 50 males and 50 females.”