1. Two Sample Hypothesis Testing (paired t-test)
PHOP 6372:
Two Sample Hypothesis Testing
(paired t-test)

2. Objective Understand how to conduct a paired t-test
Understand the relationship between a paired (dependent) t-test and a one-sample t-test

3. Examples of Paired Data
Two measurements on the same person
pretest, posttest
crossover studies
differences between left eye, right eye
Matched pairs
Differences between spouses in same couples
Key issue: the measurements are not independent.

4. Paired Sample Analysis
Equivalent to an analysis of differences
A paired-sample t-test is in fact a one-sample t-test of differences between items in the same pair.

5. Illustration 2 variables x1 and x2 with means μ1 and μ2
Example: weight gain of people on vacation
x1: weight at beginning of vacation
x2: weight at end of vacation
x2 – x1: weight gain
The mean of x2 – x1
Δ = μ2 – μ1
H0: μ1 = μ2 which is equivalent to H0: Δ = 0
H1 can be 1-tailed or 2-tailed.

6. Illustration Observation (i) x1 x2 d=x2-x1 1 x11 x12 d1=x12-x11 2 x213
x31
x32
d3=x32-x31 … …   … n
xn1
xn2
dn=xn2-xn1

7. Formula Test statistic: Thresholds: t > tn-1,1-α t < -tn-1,1-α
t > tn-1,1-α/2 or t < -tn-1,1-α/2

8. Exercise
Suppose a sample of n students were given a diagnostic test before studying a particular module and then again after completing the module. We want to find out if, in general, how our teaching affects students’ knowledge/skills (i.e. test scores). We can use the results from our sample of students to draw conclusions about the impact of the module in general.
Let’s conduct a two-sample two-sided test.
Student
Pre-module score
Post-module score
difference 1 18 22 4 2 21 25 3 16 17 . 19 15 20 -1

9. Recall Hypothesis Testing Steps
Set up null hypothesis (H0) and alternative hypothesis (H1 or HA)
Choose a test statistic
Choose significance level (), i.e. type I error rate
Determine rejection region
Reject H0 if test statistic falls in rejection region

10. Example H0: μd = 0 H1: μd ≠ 0 We hope to find evidence to reject H1.
We will look at the sample mean , the average difference in our data.
If is substantially different than 0, we will conclude that μd ≠ 0.

11. Example (cont.) H0: μd = 0 H1: μd ≠ 0 Choose α = 0.05
Then the rejection region is t > t20-1, = 1.729
Test statistic
t is in the rejection region
reject H0

12. Critical region for previous example
twoway function y=tden(19,x), range(-3 3) droplines(0, 1.729) xlabel(-3(1)3) title(Student's t-distribution with 19 df)

13. Stata outputs-data example
gen diff= postmod-premod
graph box diff
sktest diff
Reasonably normal to continue with t-test

14. Stata outputs summarize diff ttest diff=0
The two-sided p value is Therefore, there is strong evidence to reject the null hypothesis. he difference in running time between the two computers is not statistically significant.