1. When Data Comes in Pairs
t-Tests and
two-sample problems

2. Two-sample problems… Are very common in statistics
Arise when we want to compare two populations (sample and control), where:
each group is a distinct population
responses in each group are independent of those in the other group

3. Combining Means and Standard Deviations
In Chapter 4 we learned how to combine means and standard deviations. We summarize below as:
Population
Variable or sample size
Mean or Sample Mean
s or SE 1
X1,n1 2
X2,n2

4. Two Sample Statistics
Two-Sample Z
Two-Sample t

5. Example 7.14
Are the control and treatment group results different?
Solution – Key Steps
► inspect the data!
► determine key statistics for each group
► formulate a null hypothesis
► find p-value for null hypothesis
Group n s
Tmt 21
51.48
11.01
Ctrl 23
41.52
17.15

6. Carry out the two-sample t-test:
If Ho is true
How do we determine
the df for this?
Conservative approach: pick smallest n-1 = 20 for df
Use software estimation (df = 37.9)

7. Answer is… From Minitab: “By Hand”: choose df = 20 P <0.02
So – either way the null hypothesis is rejected. The treatment mean is higher.

8. Two-Sample t-Confidence Intervals…
What is the 95% (“19 times out of 20”) confidence level for the mean reading improvement?
Use
Note: t* can always be estimated conservatively by choosing the lesser of the two df’s.

9. Group Work… 7.58 7.60 7.68 n Average SD Failed 33 0.824 0.48 Healthy
1.726
0.64

10. In conclusion…
The new idea here is how to combine means and standard deviations from two populations – other than that it’s just t-tests.
Make sure you are aware of how the df changes and what the conservative approach is.
Try 7.65, 7.74, 7.81