LI Econometrics
EXERCISES CLASS 1
1. A random variable X is normally distributed with a mean of 7 and a variance of 4. Calculate
the probability that X will take on a value (a) greater than 9; (b) greater than 3; (c) less
than 8 but greater than 2.
2. Given following random sample from a normally distributed population:
5, 8, 11, 6, 15, 9
(a) Calculate the mean and variance for this sample
(b) Use both CI and test of significance approaches to test the null hypothesis that the
population mean is 11 at the 5% level of significance.
3. A random sample of size 100 is drawn from a normally distributed population of marks
from an A-level exam, with a sample mean of 68 and a variance of 8.5. Test the null
hypothesis that the average for this exam is greater than 70 at the 5% significance level.
4. (a) What is meant by a “linear estimator”? Show that the sample mean is a linear
estimator.
(b) What is meant by an “unbiased estimator”? Show that the sample mean is an unbiased
estimator.
(c) Derive the variance of the sample mean estimator.
(d) What is meant by a “best estimator”? What is the sampling distribution of the sample
mean X̄? Show that the sample mean
efficient than the sample median, which
is more
πσ 2
has sampling distribution Xm ∼ N µ, 2n
5. A random sample of size n=15 is drawn from a normal distribution with mean=11.5 and
variance=9.3. Determine the 95% and the 90% CI for the population mean µ. Interpret
your results.
Mary Dawood – University of Birmingham | Page 1