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?