Download Chapter 6.5 The Central Limit Theorem

Chapter 6.5 The Central Limit Theorem
The sampling distribution of the mean is the probability distribution of the sample
means (all the same size). As the sample size increases, the sampling distribution of the
sample means approaches a normal distribution.
We denote the mean of the sample mean by μX and the standard deviation of the sample
means by σ X which is also called the standard error of the mean.
Example:
The following is the probability distribution of rolling a single fair die.
x
1
2
3
4
5
6
p(x)
1/6
1/6
1/6
1/6
1/6
1/6
μ = Σx ⋅ p(x) = 3.5
σ = Σx 2 ⋅ p(x) − μ 2 = 35
12
Now consider the probability distribution of the mean of the sum of two dice.
Sum
2
3
4
5
6
7
8
9
10
11
12
Mean
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
p(x)
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
μX = Σx ⋅ p(x) = 3.5
σ X = Σx 2 ⋅ p(x) − μ 2 = 35
24
Properties:
•
μX = μ
σ = σ
•
z=
•
•
•
X
n
x−μ
σ
n
For sample sizes greater than 30, the sample means can be approximated by a
normal distribution.
If the original population is normal, then the sample means will be normal for any
size n.
Example:
The heights of 18–year old men are approximately normally distributed with mean
68 inches and standard deviation 3 inches.
a.
What is the probability that an 18–year old man selected at random is at
least 70 inches tall?
μ = 68
σ=3
P(x > 70)
z = 70 − 68 = .67
3
P(z > .67) = .2514
The probability that an 18–yr old man
is at least 70 in tall is 0.2514
b.
If a random sample of ten 18–year–old men is selected, what is the
probability that the mean height is at least 70 inches tall?
μ = 68
σ=3
n = 10
P( x > 70)
z = 70 − 68 = 2.11
3
10
P(z > 2.11) = .0174
The probability that a sample of 18–yr
old man will have a mean of at least 70
in tall is 0.0174