1. 366_7

2. T-distribution T-test vs. Z-test Z assumes we know, or can calculate the standard error of the distribution of something in a population We never really do

3. T-distribution Reality: We estimate standard error of a mean we observe from (duh) what we observe Insert formula for estimate of standard dev. of sample here (p. 138) & std error of mean (p. 139)

4. T-ratio t-ratio used to test if/how an observation from a sample reflects the population mean very similar to Z-scores (at infinity) Slightly different distribution

5. T-ratio Suppose observing children 10 cooperating how many cooperative acts? how confident it reflects population 1 5 2 3 4 1 2 4 31) mean = 2.7

6. T-ratio Suppose observing children 10 cooperating 2) Calculate std. deviation sum of sq distances from mean sum X 2 =89 sd=SQRT (89/10) – 2.7 =1.27

7. T-ratio Suppose observing children 10 cooperating 3) Translate std dev. into std. error s.e= s.d / SQRT(N-1) =.42 4) establish degrees of freedom & alpha (.05) df = N-1 = 9 5) Check Table C

8. T-ratio how confident? 95% confident the population mean is between 1.74 to 3.66 acts. 6) t= 2.26 7) Confidence intervals 95%= 2.7 +/-(2.62*.42) = 2.7 +/-.96 = 1.74 +/- 3.66

9. Hypothesis Testing with t Research Hypothesis (H1): Something is going on. There is a difference between groups, Men have higher score. H1: X m > X f Null Hypothesis (H0): There is no difference Mean for group 1 = the mean for group 2 H0: X1 = X2

10. Hypothesis Testing with t Observe difference between means: Magnitude of difference Variance in measure of X1 and X2 Number of observations What is the likelihood that such a difference would occur by chance?

11. T-test Assume – Random samples, independent of each other – Variable being compared is interval or ratio – Distributions are normal – Roughly equal variance of each group

12. T-test Decide criteria, or critical t – Alpha to reject (chance of a Type 1 error) t= 1.65 for alpha =.10 (at infinity. Critical t larger if sample small) t= 1.96 for alpha -.05 (larger if sample small) – Directional test?

13. t test Calculate t t = (x1 – x2) ________________ s x1-x2  ----------std. error of the difference between 2 means this part is messy, but includes info about sample sizes

14. t-test result is one value (t-statistic) we can use to check if difference between groups is significant Example: – Corruption, south vs. non south What hypothesis?

15. Projects Identify testable hypotheses – x causes y – x explains differences in y – differences in x explain y – x and y go together in some interesting way Identify how to measure what you want to test – What actual questions We’ll worry about the test statistic later

16. Mean of group 1 significantly different than mean of group 2? Non south, x=.33; s.e..03South, x=.43; s.e.05

17. t-test Southern states, zoomed in....

18. Results ttest percap_convic, by(var82) Two-sample t test with equal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- 0 | 39.3358051.0313071.1955129.2724272.3991831 1 | 11.4324917.0577252.1914528.303872.5611115 ---------+-------------------------------------------------------------------- combined | 50.3570762.0278429.1968793.3011237.4130287 ---------+-------------------------------------------------------------------- diff | -.0966866.0664606 -.2303146.0369414 ------------------------------------------------------------------------------ diff = mean(0) - mean(1) t = -1.4548 Ho: diff = 0 degrees of freedom = 48 Ha: diff 0 Pr(T |t|) = 0.1522 Pr(T > t) = 0.9239.

19. Results Note each mean is given variation around mean is given confidence intervals difference between means is given(-.096) std. error of differences btwn means given AND t values

20. Results Note different t values are given Each is for a specific hypothesis – Difference is greater than 0, positive(one tail) – Difference is greater than 0, negative(one tail) – “Absolute difference” (two tail)

21. t test results Do we accept of reject null hypothesis?