Homework 5(Dennis)
1. What does the Central Limit Theorem tell us about the sampling distribution of the sample mean?
Solution: If the sample size n is sufficiently large, then the population of all sample
means is approximately normal.
2. Suppose that we will take a random sample of size n from a population having mean  and
standard deviation  . For each of the following situations, find the mean, variance, and standard
deviation of the sampling distribution of the sample mean x :
a   10 ,   2 , n=25
b   3 ,   0.1, n=4
c   500 ,   0.5 , n=100
d   100 ,   1 , n=1,600
a.  x  10
2
22
4

.16
n
25 25

2
2
x 

 .4
n
25 5
 x2 
b.
x  3
 x2 
x 
c.

2
n


(.1) 2 .01

.0025
4
4

.1
n
4

.1
.05
2
 x  500
2
(.5) 2 .25

.0025
n
100 100

.5
.5

.1 .1
x 

 .05  x 

 .05
n
100 10
n
4 2
 x2 
d.

 x  100
2
(1) 2
1

.000625
n 1600 1600

1
1
x 


.025
n
1600 40
 x2 

3. THE BANK CUSTOMER WAITING TIME CASE
Recall that the bank manager wants to show that the new system reduces typical customer waiting
times to less than 6 minutes. One way to do this is to demonstrate that mean of the population of all
customer waiting times is less than 6. Letting this mean be  , in this exercise we wish to investigate
whether the sample of 100 waiting times provides evidence to support the claim that  is less than 6.
For the sake of argument, we will begin by assuming that  equals 6, and we will then attempt to
use the sample to contradict this assumption in favour of the conclusion that  is less than 6. Recall
that the mean of the sample of 100 waiting times is x =5.46 and assume that  , the standard deviation
of the population of all customer waiting times, is known to be 2.47.
a. Consider the population of all possible sample means obtained from random samples of 100
waiting times. What is the shape of this population of sample means? That is, what is the shape of
the sampling distribution of x ? Why is this true?
b. Find the mean and standard deviation of the population of all possible sample means when we
assume that  equals 6.
c.
The sample mean that we have actually observed is x =5.46. Assuming that  equals 6, find the
probability of observing a sample mean that is less than or equal to x =5.46.
Solution: a.
Normal because the sample is large (n  30)
b.
 x  6, x 
c.

n

2.47
100
 .247
5.46  6 

P( x  5.46)  P z 
  P( z  2.19)  1  .9857  .0143
.247 

4. The sampling distribution of
deviation 1.
FALSE
must be a normal distribution with a mean 0 and standard
5. For any sampled population, the population of all sample means is approximately normally
distributed.
FALSE
6. The sampling distribution of a sample statistic is the probability distribution of the population of all
possible values of the sample statistic.
TRUE
7. The standard deviation of all possible sample proportions increases as the sample size increases.
FALSE
8. The central limit theorem states that as sample size increases, the population distribution more
closely approximates a normal distribution.
FALSE
9. If a population is known to be normally distributed, then it follows that the sample standard
deviation must equal .
FALSE
10. The central limit theorem states that as the sample size increases the distribution of the sample
________ approach the normal distribution.
A. Medians
B. Means
C. Standard deviations
D. Variances
11. As the sample size ______________ the variation of the sampling distribution
of
___________.
A. Decreases, decreases
B. Increases, remains the same
C. Decreases, remains the same
D. Increases, decreases
12. If the sampled population has a mean 48 and standard deviation 16, then the mean and the
standard deviation for the sampling distribution of
A. 4 and 1
B. 12 and 4
C. 48 and 4
D. 48 and 1
E. 48 and 16
for n=16 are: