Download 1 Example The heights of kindergarten children are approximately

NOTES 6-5
Central Limit Theorem
Sometimes in statistics we want to know
about the probability of the SAMPLE MEAN.
To do this we need to find the mean of the
means and the standard deviation of the
means, then continue to follow the normal
distribution procedures.
The Central Limit Theorem
As the sample size n increases without limit, the shape
of the distribution of the sample means taken with
replacement from a population with mean μ and
standard deviation σ will approach a normal
distribution.
We can use the Central Limit Theorem as the
sample size n increases without limit. Large
sample sizes!!!
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Example The heights of kindergarten children are approximately normally distributed with a mean of 39 and a standard deviation of 2. If one child is randomly selected, what is the probability that the child is taller than 41 inches?
This is 1 child – Not the Central Limit Theorem!
Let's solve this.....
Apr 6­9:05 AM
Example Suppose we have a class of 30 kindergarten children. What is the probability that the mean height of these children exceeds 41 inches?
This is the Central Limit Theorem as it is asking about the probability of a sample mean!
We will wait to solve this one until later....
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Mar 20­10:19 PM
1
Properties of the Distribution of Sample
Means:
1)
Back to the Example from before:
Suppose there is a kindergarten class of 30
children. What is the probability that the
mean height exceeds 41 inches?
(Remember μ=39 and σ= 2)
**Do everything in one step to avoid rounding
errors!
2)
So really the only difference is a new formula must be used for z values:
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Conclusion:
• It is not unusual for one child, selected at random from a kindergarten class, to be taller than 41 inches. (16%)
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Example
In a certain population SAT scores are
normally distributed with mean = 500 and
standard deviation = 100. In a sample size
of 25, what is the probability that their
sample mean score is between 490 and
510?
• It is highly unlikely that the mean height for 30 kindergarten students exceeds 41 inches. (0%)
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Example
The average number of pounds of meat that a
person consumes a year is 218.4 pounds.
Assume the standard deviation is 25 pounds
and the distribution is approximately normal.
a) Find the probability that a person
selected at random consumes less than
224 pounds per year.
HW pg 325 ­ 326 8, 9, 10, 12, 17
b) If a sample of 40 individuals is selected,
find the probability that the mean of the
sample will be less than 224 pounds per
year.
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