1. Sociology 5811: T-Tests for Difference in Means
Wes Longhofer, pinch-hitting for Evan Schofer

2. Strategy for Mean Difference
We never know true population means
So, we never know true value of difference in means
So, we don’t know if groups really differ
If we can figure out the sampling distribution of the difference in means…
We can guess the range in which it typically falls
If it is improbable for the sampling distribution to overlap with zero, then the population means probably differ
An extension of the Central Limit Theorem provides information necessary to do calculations!

3. Sampling Distribution for Difference in Means
The mean (Y-bar) is a variable that changes depending on the particular sample we took
Similarly, the differences in means for two groups varies, depending on which two samples we chose
The distribution of all possible estimates of the difference in means is a sampling distribution!
The “sampling distribution of differences in means”
It reflects the full range of possible estimates of the difference in means.

4. Mean Differences for Small Samples
Sample Size: rule of thumb
Total N (of both groups) > 100 can safely be treated as “large” in most cases
Total N (of both groups) < 100 is possibly problematic
Total N (of both groups) < 60 is considered “small” in most cases
If N is small, the sampling distribution of mean difference cannot be assumed to be normal
Again, we turn to the T-distribution.

5. Mean Differences for Small Samples
To use T-tests for small samples, the following criteria must be met:
1. Both samples are randomly drawn from normally distributed populations
2. Both samples have roughly the same variance (and thus same standard deviation)
To the extent that these assumptions are violated, the T-test will become less accurate
Check histogram to verify!
But, in practice, T-tests are fairly robust.

6. Mean Differences for Small Samples
For small samples, the estimator of the Standard Error is derived from the variance of both groups (i.e. it is “pooled”)
Formulas:

7. Probabilities for Mean Difference
A T-value may be calculated:
Where (N1 + N2 – 2) refers to the number of degrees of freedom
Recall, t is a “family” of distributions
Look up t-dist for “N1 + N2 -2” degrees of freedom.

8. T-test for Mean Difference
Back to the example: 20 boys & 20 girls
Boys: Y-bar = 72.75, s = 8.80
Girls: Y-bar = 78.20, s = 9.55
Let’s do a hypothesis test to see if the means differ:
Use a-level of .05
H0: Means are the same (mboys = mgirls)
H1: Means differ (mboys ≠ mgirls).

9. T-test for Mean Difference
Calculate t-value:

10. T-Test for Mean Difference
We need to calculate the Standard Error of the difference in means:

11. T-Test for Mean Difference
We also need to calculate the Standard Error of the difference in means:

12. T-test for Mean Difference
Plugging in Values:

13. T-test for Mean Difference

14. T-Test for Mean Difference
Question: What is the critical value for a=.05, two-tailed T-test, 38 degrees of freedom (df)?
Answer: Critical Value = approx. 2.03
Observed T-value = 1.88
Can we reject the null hypothesis (H0)?
Answer: No! Not quite!
We reject when t > critical value

15. T-Test for Mean Difference
The two-tailed test hypotheses were:
Question: What hypotheses would we use for the one-tailed test?

16. T-Test for Mean Difference
Question: What is the critical value for a=.05, one-tailed T-test, 38 degrees of freedom (df)?
Answer: Around (40 df)
One-tailed test: T =1.88 > 1.684
We can reject the null hypothesis!!!
Moral of the story:
If you have strong directional suspicions ahead of time, use a one-tailed test. It increases your chances of rejecting H0.
But, it wouldn’t have made a difference at a=.01

17. Another Example
Question: Do the mean batting averages for American League and National League teams differ?
Use a random sample of teams over time
American League: Y-bar = .2677, s = .0068, N=14
National League: Y-bar = .2615, s = .0063, N=16
Let’s do a hypothesis test to see if the means differ:
Use a-level of .05
H0: Means are the same (mAmerican = mNational)
H1: Means differ (mAmerican ≠ mNational)

18. T-test for Mean Difference
Calculate t-value:

19. T-Test for Mean Difference
We need to calculate the Standard Error of the difference in means:

20. T-Test for Mean Difference
We also need to calculate the Standard Error of the difference in means:

21. T-test for Mean Difference
Plugging in Values:

22. T-test for Mean Difference

23. T-Test for Mean Difference
Question: What is the critical value for a=.05, two-tailed T-test, 28 degrees of freedom (df)?
Answer: Critical Value = approx. 2.05
Observed T-value = 2.58
Can we reject the null hypothesis (H0)?
Answer: Yes
We reject when t > critical value
What if we used an a-level of .01?
Critical value=2.76

24. T-Test for Mean Difference
Question: What if you wanted to compare 3 or more groups, instead of just two?
Example: Test scores for students in different educational tracks: honors, regular, remedial
Can you use T-tests for 3+ groups?
Answer: Sort of… You can do a T-test for every combination of groups
e.g., honors & reg, honors & remedial, reg & remedial
But, the possibility of a Type I error proliferates… 5% for each test
With 5 groups, chance of error reaches 50%
Solution: ANOVA.