Download 5.3 The Central Limit Theorem

5.3 The Central Limit Theorem
•Roll a die 5 times and record the value of each roll.
•Find the mean of the values of the 5 rolls.
•Repeat this 250 times.
x=3.504
s=.7826
n=5
•Roll a die 10 times and record the value of each
roll.
•Find the mean of the values of the 10 rolls
•Repeat this 250 times.
Poll: Toss a die 10 times and record your resu...
x=3.48
s=.5321
n=10
•Roll a die 20 times.
•Find the mean of the values of the 20 rolls.
•Repeat this 250 times.
x=3.487
s=.4155
n=20
What do you notice about
the shape of the
distribution of sample
means?
Central Limit Theorem
• Suppose we take many random samples of
size n for a variable with any distribution--For large sample sizes:
1. The distribution of means will be
approximately a normal distribution.
1, 2, 3, 4, 5, 6
• Mean: =3.5
• Standard Deviation: =1.7078
• How does the mean of the sample means
compare to the mean of the population?
• Remember for 250 trials:
• When n=5, x=3.504
• When n=10, x=3.48
• When n=20, x=3.487
• How does the mean of the sample means
compare to the mean of the population?
Central Limit Theorem
• Suppose we take many random samples of
size n for a variable with any distribution--For large sample sizes:
1. The distribution of means will be
approximately a normal distribution.
2. The mean of the distribution of means
approaches the population mean, .
1, 2, 3, 4, 5, 6
• Mean: =3.5
• Standard Deviation: =1.7078
• How does the standard deviation of the
sample means compare to the standard
deviation of the population?
• Remember for 250 trials:
• When n=5, s=.7826
• When n=10, s=.5321
• When n=20, s=.4155
• How does the standard deviation of the
sample means compare to the standard
deviation of the population?
Central Limit Theorem
• Suppose we take many random samples of
size n for a variable with any distribution--For large sample sizes:
1. The distribution of means will be
approximately a normal distribution.
2. The mean of the distribution of means
approaches the population mean, .
3. The standard deviation of the distribution

of means approaches
.
n
Cost of owning a dog
• Suppose that the average yearly cost per household
of owning dog is $186.80 with a standard deviation
of $32. Assume many samples of size n are taken
from a large population of dog owners and the mean
cost is computed for each sample.
• If the sample size is n=25, find the mean and
standard deviation of the sample means.
• If the sample size is n=100, find the mean and
standard deviation of the sample means.
Teacher’s salary
• The average teacher’s salary in New Jersey
(ranked first among states) is $52,174.
Suppose the distribution is normal with
standard deviation equal to $7500.
• What percentage of individual teachers make
less than $45,000?
• Assume a random sample of 64 teachers is
selected, what percentage of the sample
means is a salary less than $45,000?
Height of basketball players
• Assume the heights of men are normally
distributed with a mean of 70.0 inches
and a standard deviation of 2.8 inches.
• What percentage of individual men have
a height greater than 72 inches?
• The mean height of a 16 man roster on
a high school team is at least 72 inches.
What percentage of sample means from
a sample of size 16 are greater than 72
inches?
• Is this basketball team unusually tall?