Title: 7-1 and 7-2 Sampling Distribution Central Limit Theorem
17-1 and 7-2 Sampling DistributionCentral Limit
Theorem
2- Lets construct a sampling distribution (with
replacement) of size 2 from the sample set 1, 2,
3, 4, 5, 6 - 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6
- 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6
- 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6
- 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6
- 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6
- 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6
3Mean ? Probability
1 1/36
1.5 2/36
2 3/36
2.5 4/36
3 5/36
3.5 6/36
4 5/36
4.5 4/36
6 3/36
5.5 2/36
6 1/36
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5Theorem 7-1
- Some variable x has a normal distribution with
mean µ and standard deviation s - For a corresponding random sample of size n from
the x distribution - - the ? distribution will be normal,
- - the mean of the ? distribution is µ
- - the standard deviation is
6What does this mean?
- If you have a population and have the luxury of
measuring a lot of sample means, those means
(called xbar) will have a normal distribution and
those means have a mean (i.e. average value) of
mu.
7For the sample size 2
- What is the mean of 1, 2, 3, 4, 5, 6?
- What appear to be the mean of the distribution of
2 out of 6?
8Theorem 7-1 (Formula)
Why doesnt the SD stay the same? Because the
sample size is smaller you will see a smaller
deviation than you would expect for the whole
population
9Central Limit Theory
- Allows us to deal with not knowing about original
x distribution - (Central fundamental)
- The Mean of a random sample has a sampling
distribution whose shape can be approximated y
the Normal Model as the value of n increases. - Larger Sample Bigger Approximation
- The standard is that n 30.
10Example
- Coal is carried from a mine in West Virginia to a
power plant in NY in hopper cars on a long train.
The automatic hoper car loader is set to put 75
tons in each car. The actual weights of coal
loaded into each car are normally distributed
with µ 75 tons and s 0.8 tons.
11What is the probability that one car chosen at
random will have less than 74.5 tons of coal?
- This is a basic probability last chapter
12What is the probability that 20 cars chosen at
random will have a mean load weight of less than
74.5 tons?
- The question here is that the sample of 20 cars
will have ? (xbar) 74.5
13Another Example
- Invesco High Yield is a mutual fund that
specializes in high yield bonds. It has
approximately 80 or more bonds at the B or below
rating (junk bonds). Let x be a random variable
that represents the annual percentage return for
the Invesco High Yield Fund. Based on
information, x has a mean µ 10.8 and s 4.9
14- Why would it be reasonable to assume that x (the
average annual return of all bonds in the fund)
has a distribution that is approximately normal? - 80 is large enough for the Central Limit Theorem
to apply
15Compute the probability that after 5 years ? is
less than 6
- (Would that seem to indicate that µ is less than
10.8 and that the junk bond market is not
strong?)
N 5 because we are looking over 5 years
Yes. The probability that it is less than 6 is
approx. 1. If it is actually returning only 6,
then it looks like the market is weak.
16Compute the probability that after 5 years ? is
greater than 16
17Note
- The Normal model applies to quantitative data