Chapter 11: Random Sampling and Sampling Distributions
If we draw n=9 values (X) from a population that we know is normally
distributed with a mean of 100 and a standard deviation of 15 (like IQs),
what can we say about the distribution of the mean of these values?
It is not surprising that the mean of these values will be around the mean of
the population.
But what about the variability?
In general, when we’re referring to the probability distribution of a statistic,
we’re talking about that statistic’s sampling distribution. If we’re talking
about how sample means are distributed, then we’re talking about the
sampling distribution of the mean.
(See the demo)
If we repeatedly draw n samples from a normal distribution with mean m and
standard deviation s, then the mean of these samples will also be normally
distributed with a mean:
uX  uX
And standard deviation:
sX 
sX
Population Mean: 100.00, SD: 15.00
n
Means of 5000 samples of size 9: 99.87, SD 5.004
Central Limit Theorem:
The sampling distribution of the mean tends toward a normal distribution even if the
population is not normally distributed. The sampling distribution becomes more normal
for larger sample sizes. And the means and standard deviations of the sampling
distribution of the mean are still:
uX  uX
and
sX 
sX
Population Mean: 34.96, SD: 48.36
n
Means of 5000 samples: 34.97, SD 16.07
Now that we know about the
sampling distribution of the
means, we can use Table A (normal
distribution) to calculate the
probability of observing a specific
mean (or greater).
Example: Suppose the IQ of the population is distributed normally with a mean
of 100 and a standard deviation of 15. If we draw 16 people at random from
the population, what is the probability that the mean IQ of this sample will be
greater than 107?
Example: Suppose the IQ of the population is distributed normally with a mean
of 100 and a standard deviation of 15. If we draw 16 people at random from
the population, what is the probability that the mean IQ of this sample will be
greater than 107?
u X  u X  100
sX 
sX
n

The z-score for 107 is therefore
z
X u X
sx
107  100

 1.867
3.75
15
 3.75
16
Relative frequency
Answer: We know that the sampling distribution of the mean with n=16 will
have a mean and standard deviation of:
90
100
IQ
110
The area under the normal distribution above z=1.86 is 0.0314
So there is a less than 5% chance of observing a sample mean greater than 107.
Example: (IQ’s again) With a sample size of 100, what is the probability of
observing a mean IQ that is less than 99?
Example: (IQ’s again) With a sample size of 100, what is the probability of
observing a mean IQ that is less than 99?
u X  u X  100
sX 
sX
n

15
 1.5
100
The z score for a sample mean of 99 is:
z
X u X
sx
99  100

 0.667
1.5
Relative frequency
Answer: Like before:
94
96
98
100
IQ
102
104
106
The area below -0.667 is the same as the area above +0.667, which is 0.2514
So there is only a 25% chance of obtaining a sample mean more than one point
below the population mean.
Example: Suppose the height of the population of men has a mean of 70 inches and
a standard deviation of 2.8 inches. If we sample 25 men from the population, what
is the mean height that corresponds to the 95th percentile point (P95?)
Example: Suppose the height of the population of men has a mean of 70 inches and
a standard deviation of 2.8 inches. If we sample 25 men from the population, what
is the mean height that corresponds to the 95th percentile point (P95?)
Answer: The sampling distribution of the mean has a mean and standard deviation
of:
u X  u X  70
sX
2.8
sX 

 0.56
n
25
z
X u X
sx
, X  zs x  u X
X  (1.65)(0.56)  70  70.9
Relative frequency
The z-score for the upper 5% of the normal distribution is 1.65.
68
69
70
height
71
So there is a 5% chance of observing a mean of 70.9 or more inches.
72