Inference for Distributions Chapter 7 Inference for Distributions

Chapter 7 So far, we have assumed that s was known. If we assume that s is not known, we can use the sample standard deviation, s. But this adds more variability to our test statistic and/or confidence interval (therefore, we will use the t-table).

Terminology If s is known, then s /n is known as the standard deviation of x. If s is not known, then s/n is known as the standard error of x. When s is not known, we use the t-table instead of the z table.

The t-distribution Need degrees of freedom (referred to as df). In the “one sample problem” df = n-1 As df increases, the t-distribution gets closer to the standard normal. t-distribution is located in the back of the book Examples of using t-distribution.

Confidence intervals for m x + tn-1s/√n Find a 95% confidence interval for m for the following data set: 5 8 7 10 12 17 12 13 9 6 14 11 10 x = 10.308, s = 3.376

Hypothesis testing Same 5 steps Test statistic is tCALC = (x – m)/(s/√n) In previous data set, test that the mean is significantly higher than 9.5.

Matched Pairs Subjects are matched in “pairs” and outcomes are compared within each unit Examples: before scores versus after scores Twins: one gets Treatment A and the other gets Treatment B We perform hypothesis testing on the difference in each unit

Matched pairs Example: Before After 75 80 66 73 50 90 77 73 Diff 5 7 75 80 66 73 50 90 77 73 Diff 5 7 40 -4

Matched pairs Stating hypotheses: The null hypothesis of NO difference between the two groups is H0:m = 0 When stating the alternative, be careful how you are calculating the difference (after – before or before – after).

Matched Pairs If we take After – Before, and we want to show that the “After group” has increased over the “Before group” Ha: m > 0 “After group” has decreased Ha: m < 0 The two groups are different Ha: m ≠0

Matched pairs Note: The t procedure is relatively robust to nonnormal distributions (for moderate n). Example 7.8 (use your knowledge, p.431)

Chapter 7.2 Notation Sampling distribution of x1 - x2 Null hypothesis that the two groups are equal: H0: m1 = m2 Refer to Problem 7.61 (total cholesterol) Construct a 95% confidence interval for the mean difference in Total Cholesterol

The pooled two-sample t-test If we assume the two populations have the same standard deviation, then we are able to “pool” their standard deviations. Repeat the above problem using a pooled standard deviation Section 7.3 Test hypothesis of equal variances.