Statistics 5021 – Homework 3 (solutions)
There are 20 total points (1 point for each part of each question and 3 points for handing
in the assignment). One question will be graded for correctness. This homework is due
Thursday, February 17 in your lab section.
1. Deﬁne the following terms.
(a) Random sample
Solution:
a sequence of random variables
X
1
,...,X
n
that are independent and
all have the same distribution.
(b) Realization of a random sample
Solution:
the sequence of values
x
1
,...,x
n
that the random sample
X
1
,...,X
n
took on after the experiment was performed.
2. Suppose that we model the time, in minutes, it takes to drive from Apple Valley to
the University of Minnesota during rush hour with the
N
(
μ
= 44
,σ
= 8) distribution.
In particular, 86 individuals plan to make this drive. Let
X
1
,...,X
86
denote the
yettobe measured drive times. Assume that
X
1
,...,X
86
are a random sample from
N
(
μ
= 44
,σ
= 8).
(a) What is the the probability distribution of the sample mean drive time of these 86
individuals who plan to make this drive? (specify the values of its parameters).
Solution:
Since
X
1
,...,X
86
are a random sample from
N
(
μ
= 44
,σ
= 8), using
Theorem A, the mean and standard deviation of
¯
X
are
μ
¯
X
=
μ
= 44 minutes
and
σ
¯
X
=
σ/
√
n
= 8
/
√
86 = 0
.
8627 minutes. Using the ﬁrst part of Theorem B,
¯
X
∼
N
(
μ
¯
X
,σ
¯
X
).
(b) What is the name of the probability distribution of the random variable:
¯
X

44
8
/
√
86
Solution:
We know that
¯
X
is normal with mean
μ
¯
X
=
μ
= 44 and standard
deviation
σ
¯
X
=
σ/
√
n
= 8
/
√
86. Using the Normal–Standard normal rule, the
random variable:
¯
X

μ
¯
X
σ
¯
X
must be standard normal.
1