1. Sampling Distributions
Chapter 7
Central Limit Theorem
Goal: Use and interpret results using the Central Limit Theorem

2. Goal: Use and interpret results using the Central Limit Theorem
Sampling Distribution Simulation

3. Goal: Use and interpret results using the Central Limit Theorem
Take a random sample of size n from any population with mean m and standard deviation s. When n is large, the sampling distribution of the sample mean is close to the normal distribution.
How large a sample size is needed depends on the shape of the population distribution.
Rule of Thumb – N=30 will guarantee normality for all shapes of population distributions

4. Goal: Use and interpret results using the Central Limit Theorem
Uniform distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 1

5. Goal: Use and interpret results using the Central Limit Theorem
Uniform distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 2

6. Goal: Use and interpret results using the Central Limit Theorem
Uniform distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 3

7. Goal: Use and interpret results using the Central Limit Theorem
Uniform distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 4

8. Goal: Use and interpret results using the Central Limit Theorem
Uniform distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 8

9. Goal: Use and interpret results using the Central Limit Theorem
Uniform distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 16

10. Goal: Use and interpret results using the Central Limit Theorem
Uniform distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 32

11. Triangle distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 1

12. Triangle distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 2

13. Triangle distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 3

14. Triangle distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 4

15. Triangle distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 8

16. Triangle distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 16

17. Triangle distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 32

18. Goal: Use and interpret results using the Central Limit Theorem
Inverse distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 1

19. Goal: Use and interpret results using the Central Limit Theorem
Inverse distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 2

20. Goal: Use and interpret results using the Central Limit Theorem
Inverse distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 3

21. Goal: Use and interpret results using the Central Limit Theorem
Inverse distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 4

22. Goal: Use and interpret results using the Central Limit Theorem
Inverse distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 8

23. Goal: Use and interpret results using the Central Limit Theorem
Inverse distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 16

24. Goal: Use and interpret results using the Central Limit Theorem
Inverse distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 32

25. Parabolic distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 1

26. Parabolic distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 2

27. Parabolic distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 3

28. Parabolic distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 4

29. Parabolic distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 8

30. Parabolic distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 16

31. Parabolic distribution
Goal: Use and interpret results using the Central Limit Theorem
Sample size 32

32. Goal: Use and interpret results using the Central Limit Theorem
Loose ends
Goal: Use and interpret results using the Central Limit Theorem
An unbiased statistic falls sometimes above and sometimes below the actual mean, it shows no tendency to over or underestimate.
As long as the population is much larger than the sample (rule of thumb, 10 times larger), the spread of the sampling distribution is approximately the same for any size population.

33. Goal: Use and interpret results using the Central Limit Theorem
Loose ends
Goal: Use and interpret results using the Central Limit Theorem
As the sampling standard deviation continually decreases, what conclusion can we make regarding each individual sample mean with respect to the population mean m?
As the sample size increases, the mean of the observed sample gets closer and closer to m. (law of large numbers)