1. T-Tests
Lecture: Nov. 6, 2002

2. Review
So far we have learned how to test hypothesis for two types of data:
Binomial data using the binomial distribution (test statistic: p)
Comparison of a sample mean to a known population mean and variance (test statistic: z)

3. T-Tests
Z-tests require that we know the population standard deviation, which is often NOT known.
In these cases, we have to use the sample variance and standard deviation, as estimates of the population variance and standard deviation.

4. T-Tests Population standard error (sX) = s2 / n
Use sample variance (df = N – 1)
Sample standard error (sX) = s2 / n
This creates a new statistic, the t statistic.
t = X – m / sX

5. T-Tests
The only difference between a t-test and a z-test is the use of sample variance, not population variance in the formula.
The degrees of freedom for a t-test is N-1.
The greater the degrees of freedom (i.e., the greater the N), the more accurate the t-statistic represents the z-score.

6. T-tests and Distributions
Each statistic has an underlying distribution.
Binomial p = binomial distribution which changes depending on p, N, and r.
z-scores = normal distribution
t-statistics = t-distribution
Changes depending on the degrees of freedom.
Larger the degrees of freedom, the more it approximates the normal distribution.

7. T-statistics and Distributions
The t distribution is bell-shaped, but more spread out.
Greater the N, and, thus, the degrees of freedom, the less spread out the t distribution.

8. Determining Probabilities of t
Just as you look up the probability of obtaining a z score using the unit normal table, we simply look up given t-statistic using a table for the t distribution (page 693 in textbook).
However, we must also know the degrees of freedom.

9. Determining Probabilities of t
Proportion in One Tail
.05
.025
.01
.005
Proportion in Two Tails df
.10
.02 1
6.314
12.706
31.821
63.657 2
2.920
4.303
6.965
9.925 3
2.353
3.182
4.541
5.841 4
2.132
2.776
3.747
4.604

10. Hypothesis Testing with the t-test
Example: A researcher knows that in 1990 the average age at which 25-year-olds report having their first drink was 16. The researcher predicts that that ten years later, 25-year-olds would report a significantly younger first age of drinking. She recruits year-olds for her study and asks when each participant had his/her first drink.

11. Hypothesis Testing with the t-test
First she constructs her hypotheses.
H0: mage of first drink = 16
Ha: mage of first drink = 16

12. Hypothesis Testing with the t-test
Create a decision rule.
Set the alpha level; a = .05.
Find the “critical t” in on our t-table.
Because non-directional hypothesis, we must find t-value that for which that t-value or higher is < .05 using two-tails.
Must look up based on 19 degrees of freedom (N – 1).
This t-statistic is
So, if we get a t-statistic > 2.086, we can reject the null hypothesis.

13. Hypothesis Testing with the t-test
Collect data and calculate a t-statistic.
Assume researcher found that in her sample of 20 adults, the average age of first drink was 14, and the variance of 4.
The standard error of the t-statistic.
s2/n = 4/20 = .45
The t-statistic is:
X – m / sX = 14-16/.45 = -2/.45 =

14. Hypothesis Testing with the t-test
Apply the decision rule.
Is our obtained t-statistic equal to or larger than our “critical t”?
Obtained t: -4.44
Critical t:
The absolute value of our t-statistic is more extreme. In other words, there is less than a 5% probability that if the population mean is 16 that we would obtain a sample mean of 14 with a variance of 4.
Reject the null hypothesis; accept the alternative hypothesis.

15. Assumptions of the t-test
All observations must be independent.
The population distribution of scores must be normally distributed.

16. Reporting a t-test
When reporting a t-statistic you should provide the sample mean and sample standard deviation somewhere in the paper.
You should also report the t-statistic, the degrees of freedom, whether it fell within the region of rejection, and whether it was a one- or two-tailed test.

17. Reporting a t-test Example:
The 20 participants reported the age at which they had their first drink (M = 14, SD = 2). This age was significantly different from the average age of 16 reported a decade earlier, t(19) = -4.44, p < .05, two-tailed.