The document discusses different types of t-tests used to determine if the means of two samples are statistically significantly different from each other. It describes paired sample t-tests used to compare means when the same subjects are measured before and after a treatment. It also describes two-sample t-tests used to compare independent samples that may have equal or unequal variances, and whether the tests are one-tailed or two-tailed. Examples are provided of interpreting t-test output and determining if differences are statistically significant based on the t-statistic and p-values. Non-parametric alternatives like the Mann-Whitney U test are also briefly mentioned.

1. t-tests Dr Bryan Mills

2. Why?

3. When 2 sets of data with normal distributions

4. A difference is considered significant if the probability of getting that difference by random chance is very small. P value: The probability of making an error by chance Historically we use p < 0.05

5. The magnitude of the effect How Different?

6. The Spread of Data

7. n

8. Hypothesis Tests Hypothesis: A statement which can be proven false Null hypothesis HO: “ There is no difference” Alternative hypothesis (HA): “ There is a difference…” Try to “reject the null hypothesis” If the null hypothesis is false, it is likely that our alternative hypothesis is true “ False” – there is only a small probability that the results we observed could have occurred by chance

9. Yes Significant 1 in 20 P < 0.05 No Not significant P > 0.05 Reject Null Hypothesis Alpha Level

10. Types of Error Correct Decision Type II Error Beta Null is False (true difference) Type I Error Alpha Correct Decision Null is True Reject Null (assume difference) Accept Null

11. Paired Two-Sample For Means I have two sets of data, one before an experiment (change effect) one after. Are the means significantly different (2-tailed), is the first greater (1-tailed) or is there no difference (null hypothesis). i.e. is there greater variation between the two samples than within the samples For example - have students tests scores improved after a revision session, have average wages improved after a government initiative has been put in place?

12. Two-Sample Assuming Equal Variances analysis tool homoscedastic t-test - has same variance I have two sets of data from two different settings (Grades for women v grades for men, mean profit Cornish firms v mean profit Devon firms). Do they share a common parent population (are all three means the same, population, men, women). Are the means significantly different (2-tailed), is the first greater (1-tailed) or is there no difference (null hypothesis).

13. Two-Sample Assuming Unequal Variances heteroscedastic t-test - has different variances

14. Tails NOTE: It is always more difficult to demonstrate 2-tails as the part of distribution you are looking for is reduced (0.05/2). 1-Tail 0.05 2 -Tail 0.025 * two

15. Excel Output 2.006 t Critical two-tail 0.051 P(T<=t) two-tail 1.674 t Critical one-tail  You need this ! 0.026 P(T<=t) one-tail 1.995 t Stat 53.000 df 0.000 Hypothesised Mean Diff 30.000 30.000 Observations 156.742 83.381 Variance 17.597 23.241 Mean sample 2 sample 1 t-Test: Two-Sample Assuming Unequal Variances

16. Old Way T-stat is above One tailed critical - retain (reject Ho) Two tailed is below - reject (retain Ho) t Stat 1.995 t Critical one-tail 1.674 t Critical two-tail 2.006

17. 2.027 t Critical two-tail 2.02 t Critical two-tail 0.14 P(T<=t) two-tail 0.00 P(T<=t) two-tail not significant 1.69 t Critical one-tail significant difference 1.69 t Critical one-tail p>0.05 0.071 P(T<=t) one-tail p<0.05 0.001 P(T<=t) one-tail -1.50 t Stat 3.28 t Stat 37 df 38 df 0 Hypothesized Mean Difference 0 Hypothesized Mean Difference 20 20 Observations 20 20 Observations 89.34 63.78 Variance 3.84 4.77 Variance 64.53 60.38 Mean 60.12 62.27 Mean Women 2 Men 2 Men Women 60 65 60 62

18. The difference in [whatever the data represents] between sample 1 ( M = 23.241 , VAR = 83.381 ) and sample 2 ( M = 17.597, VAR = 156.742) was statistically significant, t (29) = 1.962, p < .05, one-tailed. 2.045 t Critical two-tail 0.059 P(T<=t) two-tail 1.699 t Critical one-tail 0.030 P(T<=t) one-tail 1.962 t Stat 29.000 df 0.000 Hypothesised Mean Difference -0.036 Pearson Correlation 30.000 30.000 Observations 156.742 83.381 Variance 17.597 23.241 Mean sample 2 sample 1 t-Test: Paired Two Sample for Means

19. Non-parametric alternatives Mann-Whitney U test

20. U = n 1 n 2 + n 1 (n 1 +1) – R 1 2 U=(7)(5) + (7)(8) – 30 2 U = 35 + 28 – 30 U = 33 Which is then compared to a table of critical values http://www.umes.edu/sciences/MEESProgram/ExperimentalDesign/Parametric%20versus%20Nonparametric%20Statistics.ppt R 2 = 48 R 1 = 30 n 2 = 5 n 1 = 7 9 170 6 178 12 5 163 180 11 4 165 183 10 3 168 185 8 2 173 188 7 1 175 193 Ranks of female heights Ranks of male heights Heights of females (cm) Heights of males (cm)