Sampling Distribution of a Sample Mean

1
Sampling Distribution of a Sample Mean

• Lecture 28
• Section 8.4
• Wed, Mar 5, 2008

2
The Central Limit Theorem

• Begin with a population that has mean ? and
  standard deviation ?.
• For sample size n, the sampling distribution of
  the sample mean is approximately normal if n ?
  30, with

3
The Central Limit Theorem

• The approximation gets better and better as the
  sample size gets larger and larger.
• That is, the sampling distribution morphs from
  the original distribution to the normal
  distribution.

4
The Central Limit Theorem

• For many populations, the distribution is almost
  exactly normal when n ? 10.
• For almost all populations, if n ? 30, then the
  distribution is almost exactly normal.

5
The Central Limit Theorem

• Also, if the original population is exactly
  normal, then the sampling distribution of the
  sample mean is exactly normal for any sample
  size.
• This is all summarized on pages 536 537.

6
Example

• Suppose a population consists of the numbers 6,
  12, 18.
• Using samples of size n 1, 2, 3, 4, and 5, find
  the sampling distribution of?x.
• Draw a tree diagram showing all possibilities.

7
The Tree Diagram (n 1)

• n 1

mean 6
6
mean 12
12
mean 18
18
8
The Sampling Distribution (n 1)

• The sampling distribution of?x is
• The parameters are
• ? 12
• ?2 24

?x P(?x)
6 1/3
12 1/3
18 1/3
9
The Sampling Distribution (n 1)

• The shape of the distribution

density
1/3
mean
6
8
10
12
14
16
18
10
The Sampling Distribution (n 1)

• The shape of the distribution

density
1/3
mean
6
8
10
12
14
16
18
11
The Tree Diagram (n 2)
mean
6
6
12
9
6
12
18
6
9
12
12
12
15
18
6
12
18
15
12
18
8
12
The Sampling Distribution (n 2)

• The sampling distribution of?x is
• The parameters are
• ? 12
• ?2 12

?x P( ?x)
6 1/9
9 2/9
12 3/9
15 2/9
18 1/9
13
The Sampling Distribution (n 2)

• The shape of the distribution

density
3/9
2/9
1/9
mean
6
8
10
12
14
16
18
14
The Sampling Distribution (n 2)

• The shape of the distribution

density
3/9
2/9
1/9
mean
6
8
10
12
14
16
18
15
The Tree Diagram (n 3)
mean
6
6
6
12
8
18
10
6
8
12
6
12
10
18
12
6
10
12
12
18
18
14
6
8
6
12
10
18
12
6
10
12
12
12
12
18
14
6
12
18
12
14
18
16
6
10
6
12
12
18
14
6
18
12
12
12
14
18
16
6
14
12
18
16
18
18
16
The Sampling Distribution (n 3)

• The sampling distribution of?x is
• The parameters are
• ? 12
• ?2 8

?x P(?x)
6 1/27
8 3/27
10 6/27
12 7/27
14 6/27
16 3/27
18 1/27
17
The Sampling Distribution (n 3)

• The shape of the distribution

density
9/27
6/27
3/27
mean
6
8
10
12
14
16
18
18
The Sampling Distribution (n 3)

• The shape of the distribution

density
9/27
6/27
3/27
mean
6
8
10
12
14
16
18
19
The Sampling Distribution (n 4)

• The sampling distribution of?x is
• The parameters are
• ? 12
• ?2 6

?x P(?x)
6 1/81
7.5 4/18
9 10/81
10.5 16/81
12 19/81
13.5 16/81
15 10/81
16.5 4/81
18 1/81
20
The Sampling Distribution (n 4)

• The shape of the distribution

density
20/81
16/81
12/81
8/81
4/18
mean
6
8
10
12
14
16
18
21
The Sampling Distribution (n 4)

• The shape of the distribution

density
Normal curve
20/81
16/81
12/81
8/81
4/18
mean
6
8
10
12
14
16
18
22
The Sampling Distribution (n 5)

• The sampling distribution of?x is
• The parameters are
• ? 12
• ?2 4.8

?x P(?x) P(?x)
6 1/243 0.004
7.2 5/243 0.021
8.4 15/243 0.062
9.6 30/243 0.123
10.8 45/243 0.185
12 51/243 0.210
13.2 45/243 0.185
14.4 30/243 0.123
15.6 15/243 0.062
16.8 5/243 0.021
18 1/243 0.004
23
The Sampling Distribution (n 5)

• The shape of the distribution

density
50/243
40/243
30/243
20/243
10/243
mean
6
8
10
12
14
16
18
24
The Sampling Distribution (n 5)

• The shape of the distribution

density
Normal curve
50/243
40/243
30/243
20/243
10/243
mean
6
8
10
12
14
16
18
25
Bag A vs. Bag B

• There are two bags, Bag A and Bag B.
• Each bag contains 20,000 vouchers with values
  from 10 to 60.
• Their distributions are shown on the following
  slide.

26
Bag A vs. Bag B
1000 vouchers
Bag A
10
20
30
40
60
50
1000 vouchers
Bag B
50
10
20
30
40
60
27
Bag A vs. Bag B

• Use the TI-83 to compute the mean and standard
  deviation of each population (Bag A and Bag B).

28
Bag A vs. Bag B

• The Bag A population
• ? 23.5
• ? 14.24
• The Bag B population
• ? 46.5
• ? 14.24

29
Bag A vs. Bag B

• The hypotheses
• H0 The bag is Bag A.
• H1 The bag is Bag B.
• Suppose that we sample 100 vouchers (with
  replacement).
• Decision rule Reject H0 if the average of the
  100 vouchers is more than 35.

30
Bag A vs. Bag B

• Find the sampling distribution of?x if H0 is
  true.
• Find the sampling distribution of?x if H1 is true.

31
The Two Sampling Distributions
H0
15
20
25
30
35
40
45
50
55
H1
15
20
25
30
35
40
45
50
55
32
The Two Sampling Distributions
N(23.5, 1.424)
H0
15
20
25
30
35
40
45
50
55
N(46.5, 1.424)
H1
15
20
25
30
35
40
45
50
55
33
The Two Sampling Distributions
H0
15
20
25
30
35
40
45
50
55
H1
15
20
25
30
35
40
45
50
55
34
Bag A vs. Bag B

• What is ??
• What is ??
• How reliable is this test?

35
Example

• Suppose a brand of light bulb has a mean life of
  750 hours with a standard deviation of 120 hours.
• What is the probability that 36 of these light
  bulbs would last a total of at least 26000 hours?