Chapter 9: Sampling Distributions

1. Chapter 9: Sampling Distributions

2. 9.1 Sampling Distributions • A parameter is a number that describes the population. A parameter is a fixed number, but in practice we do not know its value because we cannot examine the entire population. • A statistic is a number that describes a sample. The value of a statistic is known when we have taken a sample, but it can change from sample to sample. We often use a statistic to estimate an unknown parameter.

3. Parameters and Statistics • Expected value µ and • Standard deviation σ and s • Probability of success p and p

4. Example 9.1 • Read and be ready to share with the class.

5. Example 9.2 • Read and be ready to share with the class.

6. Example 9.3 • Read and be ready to share with the class.

7. Sampling distribution • The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

8. Example 9.4 • Read and be ready to share with the class.

9. Describing Sampling Distributions • Shape • Center • Spread • Outliers

10. Example 9.5 • Read and be prepared to discuss with the class.

11. Bias of a Statistic • Sampling distributions allow us to describe bias more precisely by speaking of the bias of the statistic rather than the bias in a sampling method. • Bias concerns the center of the sampling distribution sampling method.

12. Unbiased Statistic • A statistic used to estimate a parameter is unbiased if the mean of the sampling distribution is equal to the true value of the parameter being estimated.

13. Variability of a Statistic • The variability of a statistic is described by the spread of its sampling distribution. The spread is determined by the sampling design and the size of the sample. Larger samples give smaller spread. • As long as the population is much larger than the sample(say at least 10 times larger) the spread of the sampling distribution is approximately the same for any population size.

14. Example 9.6 • Read and be prepared to discuss with the class.

15. Bias and variability • Think of the parameter as the target and the statistic as an arrow. • If we consistently miss the target the same way, the statistic has bias. • High variability would mean that the arrows are widely scattered all around the target. • The goal is low variability and low bias.

18. 9.2 Sample Proportions • Do you believe that NPHS is the best high school in the county? • Raise your hands if you believe this is true. • p hat = # of successes/# of trials • Is this the true proportion of people in NPHS that believe this to be true?

19. The Sampling Distribution of p hat • Read this section, this section points out how p hat is a good approximation of p because the mean of p hat is p and the standard deviation of p hat is even smaller than that of p.

20. Sampling Distribution of a Sample Proportion • Choose an SRS of size n from a large population with population proportion p having some character of interest. Let p hat be the proportion of the sample having that characteristic. Then: • The mean of the sampling distribution is exactly p. • The standard deviation of the sampling proportion is

21. Rule of Thumb 1 • Use the recipe for the standard deviation of p hat only when the population us at least 10 times as large as the sample.

22. Rule of Thumb 2 • We will use the normal approximation to the sampling distribution of p hat for values of n and p that satisfy the following conditions: • n· p > 10 and n ( 1 – p) > 10

23. Example 9.7 • Read and be prepared to explain to the class.

26. Example 9.8 • Read and be prepared to explain to the class.

30. 9.3 Sample Means • Example 9.9: read and be prepared to discuss with the class.

33. The previous histograms emphasize a principle that we will make precise in this section: • Averages are less variable than individual observations. • Averages are more normal than individual observations.

34. The mean and standard deviation of Suppose that x bar is the mean of an SRS of size n drawn from a large population with mean µ and standard deviation σ. Then the mean of the sampling distribution of x bar is and its standard deviation is

35. Behaves like p-hat • The sample mean x bar is an unbiased estimator of the population mean µ. • The values of x bar are less spread out for larger samples. Their standard deviation decreases at a rate of , so you must take a sample four times as large to cut the standard deviation of x bar in half. • You should only use the equation when the population is at least ten times as large as the sample.

36. Example 9.10 • Read and be prepared to discuss with the class.

37. Sampling Distribution of a Sample Mean from a Normal Population • Draw an SRS of size n from a population that has the normal distribution with mean µ and standard deviation σ. Then the sample mean has the normal distribution with mean µ and standard deviation .

38. Example 9.11 • Read and be prepared to discuss with the class.

40. What to bring to class tomorrow… • 25 pennies • 2 nickels • 2 dimes • 1 quarter • A container to hold your change. • PLEASE BRING THIS SO THE ACTIVITY DOESN’T GO BOOM.