7-1 and 7-2 Sampling Distribution Central Limit Theorem - PowerPoint PPT Presentation

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7-1 and 7-2 Sampling Distribution Central Limit Theorem

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7-1 and 7-2 Sampling Distribution Central Limit Theorem Let s construct a sampling distribution (with replacement) of size 2 from the sample set {1, 2, 3, 4, 5, 6 ... – PowerPoint PPT presentation

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Title: 7-1 and 7-2 Sampling Distribution Central Limit Theorem


1
7-1 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

3
Mean ? 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
4
(No Transcript)
5
Theorem 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

6
What 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.

7
For 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?

8
Theorem 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
9
Central 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.

10
Example
  • 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.

11
What 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

12
What 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

13
Another 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

15
Compute 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.
16
Compute the probability that after 5 years ? is
greater than 16
17
Note
  • The Normal model applies to quantitative data
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