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?