Download Date: Section 6.5 Elementary Statistics Digital Notes The Central

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Section 6.5
Elementary Statistics
Digital Notes
The Central Limit Theorem
The Central Limit Theorem
IF
1. The random variable 'x' has a distribution with mean  and standard deviation  .
AND
2. Simple random samples all of the same size 'n' are selected from the population.
THEN
1. The distribution of sample means x will, as the sample size increases, approach a normal
distribution.
2. The mean of all sample means is the population mean  .

3. The standard deviation of all sample means is
n
Rule of Thumb
To get a decent normal distribution from the sample means, either
1.
n30
2.
n≤30 , but the original population is normally distributed
OR
Notation:
The mean of all sample means is denoted:
x .
The standard deviation of all sample means is denoted:
 x
Example – 7a. If 1 SAT score is randomly selected, find the probability that it is between 1550 and
1575.
7b. If 25 SAT scores are randomly selected, find the probability that they have a mean between 1550
and 1575.
7c. Why can the central limit theorem be used in part (b), even though the sample size does not exceed
30?
Example – 15. The Boeing 757-200 ER airliner carries 200 passengers and has doors with a height of
72 in. Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8
in.
a. If a male passenger is randomly selected, find the probability that he can fit through the doorway
without bending.
b. If half of the 200 passengers are men, find the probability that the mean height of the 100 men is
less than 72 in.
c. When considering the comfort and safety of passengers, which result is more relevent?
d. When considering the comfort and safety of passengers, why are women ignored in this case?
Using CLT (Central Limit Theorem)
If you are asked about a probability for a single individual of the population, DO NOT USE CLT.
Simple convert to a z-score like usual:
z=
x−

If you are asked about a probability concerning the mean of a group of individuals from the population,
USE CLT. Convert to a z-score like this:
z=
x− x
 x
Example – 12. See page 297
The Population
Let's find the mean and standard deviation of our sample means. This should give us a good idea of the
mean of The Population.
Why does the mean of the sample means give us a more reliable idea of the population mean, rather
than just one individual sample mean?