Homework Assignment 3
Statistics I
Applied Research Skills for Economists
ECON 775
1. Suppose that you collect the following sample of four observations, drawn
randomly from a normal population, representing the January heating cost for
a 4-bedroom house in Champaign, Illinois: 99, 115, 91, 79. Calculate the
mean of this sample.
a. What is the probability of obtaining this
x
or one smaller if the parent
population has mean
µ
=110 and unknown variance? What is the
probability that
x
is this large or larger if the population has mean
µ
=80
and unknown variance?
b. What is the probability of obtaining this
x
or one smaller if the parent
population has mean
µ
=110 and a standard deviation of
σ
=14? What is
the probability that
x
is this small or smaller if the population has mean
µ
=100 and
σ
=10?
2. a. Describe the difference between the standardized normal distribution and
the t-distribution. Under what conditions can each be used?
b. Is )0.2( ≥zP = 0.0228 greater than )0.2( ≥tP for all sample sizes? How
do you explain this fact?
3. Assume an economist estimates that the number of gallons of gasoline used
monthly by each automobile in the U.S. is a normally distributed random
variable with mean
µ
=50 and variance unknown:
a. Suppose that a sample of nine observations yields a sample variance of
s2 = 36. What is the probability that
x
is larger than 54 if
µ
=50? What
is the probability that
x
is less than 44 if
µ
=50? What is the probability
that
x
lies between 45 and 55?
b. How would your answers to the above problem change if n = 36?
4. Suppose that you are drawing samples of size n = 3 from a normal population
with a variance of 8.25. What is the probability that 22 /)1(
σ
sn − will exceed
5.99? What is the probability that s2 will be more than three times as large as
2
σ
?
