5.3 The Central Limit Theorem

1. 5.3 The Central Limit Theorem

2. 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.

3. Don’t forget: You can copy-paste this slide into other presentations, and move or resize the poll.

4. x=3.504 s=.7826 n=5

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.

6. Poll: Toss a die 10 times and record your resu... Don’t forget: You can copy-paste this slide into other presentations, and move or resize the poll.

7. x=3.48 s=.5321 n=10

8. Roll a die 20 times. • Find the mean of the values of the 20 rolls. • Repeat this 250 times.

9. Don’t forget: You can copy-paste this slide into other presentations, and move or resize the poll.

10. x=3.487 s=.4155 n=20

11. What do you notice about the shape of the distribution of sample means?

12. Central Limit Theorem • Suppose we take many random samples of size n for a variable with any distribution--- For large sample sizes: • The distribution of means will be approximately a normal distribution.

13. 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?

14. Central Limit Theorem • Suppose we take many random samples of size n for a variable with any distribution--- For large sample sizes: • The distribution of means will be approximately a normal distribution. • The mean of the distribution of means approaches the population mean, .

15. 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?

16. Central Limit Theorem • Suppose we take many random samples of size n for a variable with any distribution--- For large sample sizes: • The distribution of means will be approximately a normal distribution. • The mean of the distribution of means approaches the population mean, . • The standard deviation of the distribution of means approaches .

17. 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.

18. 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?

19. 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?