Docstoc

Chapter 7 The Central Limit Theorem

Document Sample
Chapter 7 The Central Limit Theorem Powered By Docstoc
					                      7.1   Sampling Distributions (Page 1 of 10)


7.1 Sampling Distributions

Parameters versus Statistics
A parameter is a
numerical measure of a      Measure               Statistic Parameter
population. A statistic is Mean                        x          
a numerical measure of a Variance                     s2          2
sample. For a given         Standard Deviation         s          
population, a parameter is Proportion                  p          
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             f
                              f/n
   Class                   Relative
              Frequency
Boundaries                Frequency
 8.39-8.86       1           0.01
 8.77-9.14       5           0.05
 9.15-9.12       10          0.10
 9.53-9.90       19          0.19
9.91-10.28       27          0.27
10.29-10.66      18          0.18
10.67-11.04      12          0.12
11.05-11.42      5           0.05
11.43-11.80      3           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 x-
distribution. 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       and        x   / n

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                        QuickTime™ and a
                                                    TIFF (Uncompressed) decompressor
                                                       are needed to see this picture.

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?




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    x         P(96  x  104) P(x  108)
                         1    100 14              0.2249
                         5    100 6.2610          0.4771
                         10   100
                         15   100
                         20   100
                         30   100
                   7.2 The Central Limit Theorem (Page 9 of 10)




  How the x -                       n = 30
  distribution
changes as the
 Sample Size
   Increases
                         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                       n    x      P(96  x  104)   P(x  108)
   increases what happens to:                 1     14         0.2249         0.2839
   i.  x                                     5   6.2610       0.4771         0.1007
                                             10   4.4272       0.6337         0.0354
                                             15   3.6148       0.7315         0.0134
   ii.  x                                   20
                                             30
                                                  3.1305
                                                  2.5560
                                                               0.7987
                                                               0.8824
                                                                              0.0053
                                                                              0.0009

   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)?
                    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 xi (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).

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:14
posted:11/26/2011
language:English
pages:10