The document discusses sampling distributions and their properties. It defines sampling error and how to calculate it. It explains that the sampling distribution of the sample mean x is normally distributed with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size. Similarly, the sampling distribution of the sample proportion p is normally distributed when the sample size is large. The Central Limit Theorem states that the sampling distribution will be approximately normal for large sample sizes regardless of the population distribution.

1. Sampling Distributions

6. Example If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is

10. Developing a Sampling Distribution .3 .2 .1 0 18 20 22 24 A B C D Uniform Distribution P(x) x (continued) Summary Measures for the Population Distribution:

11. Now consider all possible samples of size n=2 16 possible samples (sampling with replacement) (continued) Developing a Sampling Distribution 16 Sample Means

12. Sampling Distribution of All Sample Means 18 19 20 21 22 23 24 0 .1 .2 .3 P(x) x Sample Means Distribution 16 Sample Means _ Developing a Sampling Distribution (continued) (no longer uniform)

13. Summary Measures of this Sampling Distribution: Developing a Sampling Distribution (continued)

14. Comparing the Population with its Sampling Distribution 18 19 20 21 22 23 24 0 .1 .2 .3 P(x) x 18 20 22 24 A B C D 0 .1 .2 .3 Population N = 4 P(x) x _ Sample Means Distribution n = 2

15. If the Population is Normal (THEOREM 6-1) If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and

21. Central Limit Theorem n↑ As the sample size gets large enough… the sampling distribution becomes almost normal regardless of shape of population

22. If the Population is not Normal Population Distribution Sampling Distribution (becomes normal as n increases) Central Tendency Variation (Sampling with replacement) Larger sample size Smaller sample size (continued) Sampling distribution properties: