Download Chapter 7: The Central Limit Theorem

7.1
Sampling Distributions (Page 1 of 10)
7.1 Sampling Distributions
Parameters versus Statistics
A parameter is a
Measure
Statistic Parameter
numerical measure of a

x
population. A statistic is Mean
2
s2
a numerical measure of a Variance
Standard Deviation
s

sample. For a given

p
population, a parameter is Proportion
fixed, while the value of a
statistic may vary from sample to sample. Statistics estimate the
value of population parameters and provide a basis for us to make
inferences about the parameters.
Types of Inferences
1. Estimation: In this type of inference, we estimate the value
of a population parameter.
2. Testing: In this type of inference, we formulate a decision
about the value of a population parameter.
3. Regression:
In this type of inference we make
predictions or forecasts about the value of a statistical
variable.
Sampling Distribution
A sampling distribution is a probability distribution of a sample
statistic based on all possible simple random samples of the same
size from the same population.
7.1
Sampling Distributions (Page 2 of 10)
Example 1 – Sampling Distribution for x
The Pinedale Children’s fishing pond has a five fish limit. Every
child catches the limit. The data in table below is the length of each
fish caught by 100 randomly selected children. The mean x of the
5-fish sample for each child is also tabulated.
7.1
Sampling Distributions (Page 3 of 10)
a. What makes up the members of the sample?
There are 100 members in the sample - the 100 means ( x ’s) of
the 5-fish samples.
b. What is the sample statistic corresponding to each sample?
The sample statistic of each sample is the sample mean, x .
c. What population parameter is being estimated by the sample
statistic x . What population parameter are we are trying to
make an inference about?
The population mean,  .
d. What is the sampling distribution?
A sampling distribution is a probability distribution for a
sample statistic (in this case x ). It is given in table below and
its graph is given in Figure 7-1.
x
Class
Boundaries
8.39-8.86
8.77-9.14
9.15-9.12
9.53-9.90
9.91-10.28
10.29-10.66
10.67-11.04
11.05-11.42
11.43-11.80
f
Frequency
1
5
10
19
27
18
12
5
3
f/n
Relative
Frequency
0.01
0.05
0.10
0.19
0.27
0.18
0.12
0.05
0.03
7.2 The Central Limit Theorem (Page 4 of 10)
7.2 Central Limit Theorem (Part 1)
Let x be a random variable with a normal distribution whose mean
is  and standard deviation is  . Let x be the sample mean
corresponding to a random sample of size n taken from the xdistribution. Then the following are true:
(a) The x -distribution is a normal distribution.
(b) The mean of the x -distribution is the population mean (  ).
(c) The standard error is the standard deviation of the x distribution which is  x   / n .
Thus, If x has a normal distribution, then the x -distribution is a
normal distribution for any sample size n and
x   / n
x  
and
Example 2
Suppose a team of biologists has been studying the Pinedale
Children’s fishing pond of example 1 in section 7.1. Let x
represent the length of a single trout taken at random from the
pond. Assume x has a normal distribution with a  = 10.2 in. and
standard deviation  = 1.4 in.
(a) What is the probability that a single trout
taken at random from the pond is between
8 and 12 inches?
(b) What is the probability that the mean
length x of 5 trout taken at random
is between 8 and 12 inches?
(c)
Explain the difference between (a) and
(b).
7.2 The Central Limit Theorem (Page 5 of 10)
Central Limit Theorem (Part 2)
If x is any distribution with mean  and standard deviation  ,
then, as n increases without limit, the sample mean x , based on a
random sample of size n, will have a distribution that approaches a
normal distribution with mean of  x   and standard deviation
(standard error) of  x   / n .
That is, no matter what the original distribution on x, as the sample
size n gets larger and larger, the distribution of sample means
( x ’s) will approach a normal distribution.
Empirically, it has been found that in the vast majority of cases
picking a sample size of thirty or more 4( n  30 ) will yield an x distribution that is approximately normal. The larger n gets, the
more normal the x -distribution will become. Thus, for any sample
( n  30 ) on any x-distribution, we have the following
approximations:  x   , and  x   / n .
Guided Exercise 2
(a) Suppose x has a normal distribution with mean of 18 and
standard deviation of 3. If we draw random samples of size 5
from the x-distribution and x represents the sample mean,
what can be said about the x -distribution?
(b) Suppose the x distribution has mean of 75 and standard
deviation of 12. If we draw random samples of size 30 from
the x-distribution and x represents the sample mean, what can
be said about the x -distribution?
(c) Suppose you did not know if x had a normal distribution.
Would you be justified in saying that the x -distribution is
approximately normal if the sample size n = 8?
7.2 The Central Limit Theorem (Page 6 of 10)
Example 3
A strain of bacteria occurs in all raw milk. Let x be the bacteria
count per milliliter of milk. The health department has found that
if the milk is not contaminated, then the x-distribution is
approximately normal. The mean of the x-distribution is 2500, and
the standard deviation is 300. In a large dairy an inspector takes 42
random samples of milk each day and averages the bacteria count
of the 42 samples to obtain x .
(a) Assuming the milk is not contaminated, what is (describe) the
distribution of x ?
(b) Assuming the milk is not contaminated, what is the probability
that the average bacteria count for one day is between 2350
and 2650 bacteria per milliliter?
(c) Suppose one day the x for 42 samples was not between 2350
and 2650. Would you pass the sample? Why/Why Not?
7.2 The Central Limit Theorem (Page 7 of 10)
Guided Exercise 3
Tunnels are often used instead of long
roads over high passes. However, too
many vehicles in a tunnel at the same
time can be hazardous due to emissions.
If x represents the time for a vehicle to
go through a tunnel, it is known that the
x-distribution is normal with a mean of
12.1 minutes and a standard deviation of 3.8 minutes. Safety
Engineers have said that vehicles should spend between 11 and 13
minutes in the tunnel. Under ordinary conditions about 50 vehicles
are in the tunnel at a time.
a. What is the probability that the time for one vehicle (x) to pass
through the tunnel is between 11 and 13 minutes?
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
b. What is the probability that the mean time for 50 vehicles ( x )
to pass through the tunnel will be between 11 and 13 minutes?
c. Comment on the results.
7.2 The Central Limit Theorem (Page 8 of 10)
Example A
Suppose x = a person’s I.Q. score. The x-distribution is normal
with a mean of 100 and standard deviation of 14.
a. What is the probability that a person selected at random has an
I.Q. (x) between 96 and 104?
b. If a random sample of size n = 5 is drawn, find the probability
that the mean ( x ) of the sample will be between 96 and 104.
c. Repeat part b for n = 10, 15, 20, and 30.
n
x
1
5
10
15
20
30
100 14
100 6.2610
100
100
100
100
x
P(96  x  104) P(x  108)
0.2249
0.4771
7.2 The Central Limit Theorem (Page 9 of 10)
How the x distribution
changes as the
Sample Size
Increases
n = 30
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
n=5
n=1
x = IQ score
x = mean IQ score
for sample size n
d. As the sample size n
increases what happens to:
i.  x
ii.  x
iii. The probability of an x
occurring in an interval
containing  and near  .
iv. The probability of an x
occurring in an interval
not containing  and
away from  (i.e. in the
tails)?
n
1
5
10
15
20
30
x
14
6.2610
4.4272
3.6148
3.1305
2.5560
P(96  x  104)
P(x  108)
0.2249
0.4771
0.6337
0.7315
0.7987
0.8824
0.2839
0.1007
0.0354
0.0134
0.0053
0.0009
7.2 The Central Limit Theorem (Page 10 of 10)
Exercise 7.2 #17
Let x be a random variable that represents the checkout time in
minutes at the express line at a grocery store. Based on extensive
surveys the mean of the x-distribution is 2.7 minutes with a
standard deviation of 0.6 minutes. What is the probability that the
total checkout time of the next 30 customers is less than 90
minutes? Answer the question in the following steps.
a. Let x i (i = 1, 2, 3, . . . , 30) be the checkout times for the next
30 customers and w  x1  x2  x3   x30 . Explain why we
are being asked to compute P(w < 90).
b. Show that w < 90 is equivalent to x = (w/30) < 3.
c. What three things does the central limit theorem say about the
x -distribution?
d. Compute P( x < 3) = P(w < 90).