The Scientific Study of Politics (POL 51) Professor B. Jones University of California, Davis

Hypothesis Testing H_0: Statistic ``no different’’ from hypothesized value. Statistic could be: the mean or difference in means between two groups. Hypothesized value could be anything. Often, hypothetical value is 0. Why?!!

The t-test X-bar is the mean. mu is the hypothetical value. s is the sample standard deviation. Sqrt(N) is the square root of the sample size. t is the “test statistic.” If mu is 0, what are we testing? If mu is 80, what are we testing?

Alternatives Three possible alternatives to the null. Our test statistic (mean) could be: –1. Either greater or less than mu. –2. Greater than mu. –3. Lesser than mu. Statement 1: NONDIRECTIONAL Statements 2-3: DIRECTIONAL WHICH IS MORE PRECISE?

Two Ways to Do the Same Thing CONFIDENCE INTERVAL APPROACH Compute the p percent confidence interval around the statistic. Often, this is the 95 percent c.i. Test example (the second one): mu=80 Mean=84.61; s=9.71; n=31 SEM: 9.71/sqrt(31)= % CI: /-[2.04*1.74] GIVES: (81.06, 88.16) QUESTION: DOES mu=80 FALL IN THIS INTERVAL? NO!!!

Confidence Interval Approach IMPLICATIONS: The confidence interval around are test-statistic suggests that our estimate of significantly departs from 80, We conclude this at the 95 percent confidence level. That is, in repeated samples (if I kept giving this exam over and over again to this class!), 95% of all samples would produce a c.i. like this one. SINCE THIS INTERVAL DOES NOT CONTAIN mu, I AM 95% CONFIDENT IN MY DECISION TO REJECT THE NULL HYPOTHESIS. PUT DIFFERENTLY, ONLY ABOUT 5% OF REPEATED SAMPLES WOULD PRODUCE A C.I. CONTAINING mu=80. “About 5% of the time I’d be wrong.”

The t-test approach Remember my confidence interval? 95% CI: /-[2.04*1.74] Where did 2.04 come from??? We have 31 observations Therefore we have 30 “degrees of freedom” “We used 1 up for the mean”

Treasure Hunt OPEN YOUR BOOK TO P. 479 Move your finger down to 30 d.f. Move your finger to the RIGHT 2 Columns What do you see? 2.04 Move your finger up to the top of this column. What do you see? ALPHA levels. When alpha=.05 (2-tail), the CRITICAL t=2.04. ON 30 DEGREES of FREEDOM, the CENTRAL 95 PERCENT REGION of the t-DISTRIBUTION IS CONTAINED BETWEEN and +2.04

DFProbability, p

The t-test approach ALL WE REALLY NEED TO KNOW IS t. TRUTH NOW REVEALED: If the t statistic from our test meets or exceeds the critical t for alpha=.05 (or whatever you want alpha to be equal to), we REJECT the NULL HYPOTHESIS at the p percent level. If alpha=.05, our confidence level is.95 or 95 percent. If the t statistic from our test is less than the critical t we FAIL to REJECT the NULL. The t-test approach gives same conclusion as c.i. approach.

Only a few moving parts YOU DECIDE ALPHA YOU DECIDE 1-Tail or 2-Tail Examples (pretend we have 10 d.f.): t statistic is |-1.90|=1.90 Why is absolute value OK?

Mechanics t=1.90 Reject null if t >= critical t Question: Would a t statistic like this one permit us to reject the null hypothesis? Scenario 1: alpha=.05, two-tail –What is critical t? (You must have your book) –What is our decision (reject/accept null)? Scenario 2: alpha=.05, one-tail –What is critical t? –What is our decision (reject/accept null)? Scenario 3: alpha=.10, two-tail –What is critical t? –What is our decision (reject/accept null)?

ALL THIS HOLDS FOR TWO- SAMPLE t State Hypotheses Compute t Determine alpha Is t>=critical t? If yes: reject null at the (1-alpha)% level. If no: fail to reject null at the (1-alpha)% level.

Difference between 1- and 2-Tail Tests It must now be OBVIOUS that 1-tail tests are more “liberal.” The critical t is always smaller for the same alpha level in a 1-tail test vs. 2-tail test. Critical t for alpha=.05, two-tail, 10 d.f.=2.23 Critical t for alpha=.05, one-tail, 10 d.f.=1.81 You tell me: Which test makes it easier to reject the null? WHEN DO YOU CHOOSE 1 vs. 2? WHEN YOU SPECIFY DIRECTIONAL HYPOTHESIS, THEN AND ONLY THEN DO YOU CHOOSE 1-TAIL. IF YOU CANNOT SPECIFY DIRECTION, IT IS DISHONEST TO USE 1-TAIL TEST (WHY???)