1. Agresti/Franklin Statistics, 1e, 1 of 139  Section 6.4 How Likely Are the Possible Values of a Statistic? The Sampling Distribution

2. Agresti/Franklin Statistics, 1e, 2 of 139 Statistic Recall: A statistic is a numerical summary of sample data, such as a sample proportion or a sample mean.

3. Agresti/Franklin Statistics, 1e, 3 of 139 Parameter Recall: A parameter is a numerical summary of a population, such as a population proportion or a population mean.

4. Agresti/Franklin Statistics, 1e, 4 of 139 Statistics and Parameters In practice, we seldom know the values of parameters. Parameters are estimated using sample data. We use statistics to estimate parameters.

5. Agresti/Franklin Statistics, 1e, 5 of 139 Example: 2003 California Recall Election Prior to counting the votes, the proportion in favor of recalling Governor Gray Davis was an unknown parameter. An exit poll of 3160 voters reported that the sample proportion in favor of a recall was 0.54.

6. Agresti/Franklin Statistics, 1e, 6 of 139 Example: 2003 California Recall Election If a different random sample of about 3000 voters were selected, a different sample proportion would occur.

7. Agresti/Franklin Statistics, 1e, 7 of 139 Example: 2003 California Recall Election Imagine all the distinct samples of 3000 voters you could possibly get. Each such sample has a value for the sample proportion.

8. Agresti/Franklin Statistics, 1e, 8 of 139 Statistics and Parameters How do we know that a sample statistic is a good estimate of a population parameter? To answer this, we need to look at a probability distribution called the sampling distribution.

9. Agresti/Franklin Statistics, 1e, 9 of 139 Sampling Distribution The sampling distribution of a statistic is the probability distribution that specifies probabilities for the possible values the statistic can take.

10. Agresti/Franklin Statistics, 1e, 10 of 139 The Sampling Distribution of the Sample Proportion Look at each possible sample. Find the sample proportion for each sample. Construct the frequency distribution of the sample proportion values. This frequency distribution is the sampling distribution of the sample proportion.

11. Agresti/Franklin Statistics, 1e, 11 of 139 Example: Sampling Distribution Which Brand of Pizza Do You Prefer? Two Choices: A or D. Assume that half of the population prefers Brand A and half prefers Random D. Take a random sample of n = 3 tasters.

12. Agresti/Franklin Statistics, 1e, 12 of 139 Example: Sampling Distribution SampleNo. Prefer Pizza A Proportion (A,A,A) 3 1 (A,A,D) 2 2/3 (A,D,A) 2 2/3 (D,A,A) 2 2/3 (A,D,D) 1 1/3 (D,A,D) 1 1/3 (D,D,A) 1 1/3 (D,D,D) 0 0

13. Agresti/Franklin Statistics, 1e, 13 of 139 Example: Sampling Distribution Sample Proportion Probability 0 1/8 1/3 3/8 2/3 3/8 1 1/8

14. Agresti/Franklin Statistics, 1e, 14 of 139 Example: Sampling Distribution

15. Agresti/Franklin Statistics, 1e, 15 of 139 Mean and Standard Deviation of the Sampling Distribution of a Proportion For a binomial random variable with n trials and probability p of success for each, the sampling distribution of the proportion of successes has: To obtain these value, take the mean np and standard deviation for the binomial distribution of the number of successes and divide by n.

16. Agresti/Franklin Statistics, 1e, 16 of 139 Example: 2003 California Recall Election Sample: Exit poll of 3160 voters. Suppose that exactly 50% of the population of all voters voted in favor of the recall.

17. Agresti/Franklin Statistics, 1e, 17 of 139 Example: 2003 California Recall Election Describe the mean and standard deviation of the sampling distribution of the number in the sample who voted in favor of the recall. µ = np = 3160(0.50) = 1580

18. Agresti/Franklin Statistics, 1e, 18 of 139 Example: 2003 California Recall Election Describe the mean and standard deviation of the sampling distribution of the proportion in the sample who voted in favor of the recall. BE VERY CAREFUL

19. Agresti/Franklin Statistics, 1e, 19 of 139 The Standard Error To distinguish the standard deviation of a sampling distribution from the standard deviation of an ordinary probability distribution, we refer to it as a standard error.

20. Agresti/Franklin Statistics, 1e, 20 of 139 Example: 2003 California Recall Election If the population proportion supporting recall was 0.50, would it have been unlikely to observe the exit-poll sample proportion of 0.54? Based on your answer, would you be willing to predict that Davis would be recalled from office?

21. Agresti/Franklin Statistics, 1e, 21 of 139 Example: 2003 California Recall Election Fact: The sampling distribution of the sample proportion has a bell-shape with a mean µ = 0.50 and a standard deviation σ = 0.0089.

22. Agresti/Franklin Statistics, 1e, 22 of 139 Example: 2003 California Recall Election Convert the sample proportion value of 0.54 to a z-score:

23. Agresti/Franklin Statistics, 1e, 23 of 139 Example: 2003 California Recall Election

24. Agresti/Franklin Statistics, 1e, 24 of 139 Example: 2003 California Recall Election The sample proportion of 0.54 is more than four standard errors from the expected value of 0.50. The sample proportion of 0.54 voting for recall would be very unlikely if the population support were p = 0.50.

25. Agresti/Franklin Statistics, 1e, 25 of 139 Example: 2003 California Recall Election A sample proportion of 0.54 would be even more unlikely if the population support were less than 0.50. We there have strong evidence that the population support was larger than 0.50. The exit poll gives strong evidence that Governor Davis would be recalled.

26. Agresti/Franklin Statistics, 1e, 26 of 139 Example: 2003 California Recall Election Describe the mean and standard deviation of the sampling distribution of the proportion in the sample who voted in favor of the recall. BE VERY CAREFUL

27. Agresti/Franklin Statistics, 1e, 27 of 139 Summary of the Sampling Distribution of a Proportion For a random sample of size n from a population with proportion p, the sampling distribution of the sample proportion has If n is sufficiently large such that the expected numbers of outcomes of the two types, np and n(1- p), are both at least 15, then this sampling distribution has a bell-shape.

28. Agresti/Franklin Statistics, 1e, 28 of 139  Section 6.5 How Close Are Sample Means to Population Means?

29. Agresti/Franklin Statistics, 1e, 29 of 139 The Sampling Distribution of the Sample Mean The sample mean, x, is a random variable. The sample mean varies from sample to sample. By contrast, the population mean, µ, is a single fixed number.

30. Agresti/Franklin Statistics, 1e, 30 of 139 Mean and Standard Error of the Sampling Distribution of the Sample Mean For a random sample of size n from a population having mean µ and standard deviation σ, the sampling distribution of the sample mean has: Center described by the mean µ (the same as the mean of the population). Spread described by the standard error, which equals the population standard deviation divided by the square root of the sample size:

31. Agresti/Franklin Statistics, 1e, 31 of 139 Example: How Much Do Mean Sales Vary From Week to Week? Daily sales at a pizza restaurant vary from day to day. The sales figures fluctuate around a mean µ = $900 with a standard deviation σ = $300.

32. Agresti/Franklin Statistics, 1e, 32 of 139 Example: How Much Do Mean Sales Vary From Week to Week? The mean sales for the seven days in a week are computed each week. The weekly means are plotted over time. These weekly means form a sampling distribution.

33. Agresti/Franklin Statistics, 1e, 33 of 139 Example: How Much Do Mean Sales Vary From Week to Week? What are the center and spread of the sampling distribution?

34. Agresti/Franklin Statistics, 1e, 34 of 139 Sampling Distribution vs. Population Distribution

35. Agresti/Franklin Statistics, 1e, 35 of 139 Standard Error Knowing how to find a standard error gives us a mechanism for understanding how much variability to expect in sample statistics “just by chance.”

36. Agresti/Franklin Statistics, 1e, 36 of 139 Standard Error The standard error of the sample mean: As the sample size n increases, the denominator increase, so the standard error decreases. With larger samples, the sample mean is more likely to fall close to the population mean.

37. Agresti/Franklin Statistics, 1e, 37 of 139 Central Limit Theorem Question: How does the sampling distribution of the sample mean relate with respect to shape, center, and spread to the probability distribution from which the samples were taken?

38. Agresti/Franklin Statistics, 1e, 38 of 139 Central Limit Theorem For random sampling with a large sample size n, the sampling distribution of the sample mean is approximately a normal distribution. This result applies no matter what the shape of the probability distribution from which the samples are taken.

39. Agresti/Franklin Statistics, 1e, 39 of 139 Central Limit Theorem: How Large a Sample? The sampling distribution of the sample mean takes more of a bell shape as the random sample size n increases. The more skewed the population distribution, the larger n must be before the shape of the sampling distribution is close to normal. In practice, the sampling distribution is usually close to normal when the sample size n is at least about 30.

40. Agresti/Franklin Statistics, 1e, 40 of 139 A Normal Population Distribution and the Sampling Distribution If the population distribution is approximately normal, then the sampling distribution is approximately normal for all sample sizes.

41. Agresti/Franklin Statistics, 1e, 41 of 139 How Does the Central Limit Theorem Help Us Make Inferences For large n, the sampling distribution is approximately normal even if the population distribution is not. This enables us to make inferences about population means regardless of the shape of the population distribution.