Confidence Intervals and Hypothesis Testing in Statistics

1. Sections 9.1 – 9.3

2. If we want to compute a confidence interval (CI) for a population mean, which formula would we use?

3. If we want to compute a confidence interval (CI) for a population mean, which formula would we use? It depends! Think!!

4. If we know the population standard deviation, σ, we would use

8. Suppose σ is unknown What changes would decrease the width of a confidence interval (CI) for a population mean?

9. Suppose σ is unknown

10. Suppose σ is unknown

11. If we want to compute the sample size needed for a given confidence level and given margin of error, how would we proceed?

12. If we want to compute the sample size needed for a given confidence level and given margin of error, how would we proceed? Think!!! 2 possible cases

13. If we want to compute the sample size needed for a given confidence level and given margin of error, how would we proceed? Is population standard deviation, σ: • known • or unknown?

16. What does P-value mean?

17. P-value is the probability of getting a result as extreme or more extreme than the result (test statistic) we got from our sample, given the null hypothesis is true.

18. What is α?

19. α is the level of significance.

20. α is the level of significance. How does α relate to P-value?

21. α is the level of significance. α is the maximum P-value for which the null hypothesis will be rejected.

22. α is the level of significance. α is the maximum P-value for which the null hypothesis will be rejected. Reject null hypothesis because the P-value of 0.## is less than the significance level of α = 0.05

23. How does α compare for a two-sided test versus a one-sided test?

24. How does α compare for a two-sided test versus a one-sided test? It’s the same for both. For example, for a 95% confidence level, α is 0.05 for a two-sided test and a one-sided test.

25. Remember Errors? What is: • a Type I error? (b) a Type II error?

26. Errors Type I error is rejecting a true null hypothesis. (b) a Type II error?

27. Errors Type I error is rejecting a true null hypothesis. Type II error is failing to reject a false null hypothesis.

28. Errors Type I error is rejecting a true null hypothesis. P(Type I error) = ?

29. Errors Type I error is rejecting a true null hypothesis. P(Type I error) = α, the level of significance

30. Why do we transform data?

31. Why do we transform data? To change skewed data into more normal data.

32. 15/40 Guideline?

33. 15/40 Guideline? 15/40 guideline is a set of rules that helps us know when it is appropriate to use a t-interval or t-test for the population mean.

34. 15/40 Guideline Page 608

35. If our sample size is 40 or more, do we need to plot the sample data?

36. If our sample size is 40 or more, do we need to plot the sample data? Yes!! Why?

37. If our sample size is 40 or more, do we need to plot the sample data? Yes!! Why? Need to check for outliers.

38. Page 610, E41 Pretend that each data set described is a random sample and that you want to do a significance test or construct a confidence interval for the unknown mean. Use the sample size and the shape of the distribution to decide which of these descriptions (I–IV) best fits each data set

39. Page 610, E41 • There are no outliers, and there is no evidence of skewness. Methods based on the normal distribution are suitable. • The distribution is not symmetric, but the sample is large enough that it is reasonable to rely on the robustness of the t-procedure and construct a confidence interval, without transforming the data to a new scale

40. Page 610, E41 III. The shape suggests transforming. With a larger sample, this might not be necessary, but for a skewed sample of this size transforming is worth trying. IV. It would be a good idea to analyze this data set twice, once with the outliers and once without.

41. Page 610, E41 • weights, in ounces, of bags of potato chips

42. Page 610, E41 • weights, in ounces, of bags of potato chips n = 15; fairly symmetric with outlier

43. Page 610, E41 • weights, in ounces, of bags of potato chips n = 15; fairly symmetric with outlier IV. It would be a good idea to analyze this data set twice, once with the outliers and once without.

44. Page 610, E41 B. Per capita gross domestic product (GNP) for various countries

45. Page 610, E41 B. Per capita gross domestic product (GNP) for various countries n = 34, strongly skewed right

46. Page 610, E41 B. Per capita gross domestic product (GNP) for various countries n = 34, strongly skewed right III. The shape suggests transforming. With a larger sample, this might not be necessary, but for a skewed sample of this size transforming is worth trying.

47. Page 610, E41 • Batting averages of American League players

48. Page 610, E41 • Batting averages of American League players n > 40, fairly symmetric, no outliers

49. Page 610, E41 • Batting averages of American League players n > 40, fairly symmetric, no outliers I. There are no outliers, and there is no evidence of skewness. Methods based on the normal distribution are suitable.

50. Page 610, E41 D. self-reported grade-point averages of 67 students