1. Inference for Distributions
Chapter 7
Inference for Distributions

2. 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).

3. 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.

4. 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.

5. Confidence intervals for m
x + tn-1s/√n
Find a 95% confidence interval for m for the following data set:
x = , s = 3.376

6. 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.

7. 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

8. 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

9. 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).

10. 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

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

12. 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

13. 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.