1. t-test Testing Inferences about Population Means

2. Learning Objectives Compute by hand and interpret Single sample t Independent samples t Dependent samples t Use SPSS to compute the same tests and interpret the output

3. Review 6 Steps for Significance Testing 1. Set alpha (p level). 2. State hypotheses, Null and Alternative. 3. Calculate the test statistic (sample value). 4. Find the critical value of the statistic. 5. State the decision rule. 6. State the conclusion.

4. One Sample Exercise (1) 1. Set alpha.  =.05 2. State hypotheses. Null hypothesis is H 0 :  = 1000. Alternative hypothesis is H 1 :   1000. 3. Calculate the test statistic Testing whether light bulbs have a life of 1000 hours

5. Calculating the Single Sample t 800 750 940 970 790 980 820 760 1000 860 What is the mean of our sample? = 867 What is the standard deviation for our sample of light bulbs? SD= 96.73

6. Determining Significance 4. Determine the critical value. Look up in the table (Heiman, p. 708). Looking for alpha =.05, two tails with df = 10-1 = 9. Table says 2.262. 5. State decision rule. If absolute value of sample is greater than critical value, reject null. If |-4.35| > |2.262|, reject H 0.

7. α(1 tail)0.050.0250.050.0250.050.0250.050.0250.050.025 α(2 tail)0.100.0500.100.0500.100.0500.100.0500.100.050 df 16.313812.707211.72072.0796411.68292.0196611.67021.9996811.66391.9897 22.92004.3026221.71722.0739421.68202.0181621.66981.9990821.66361.9893 32.35343.1824231.71392.0686431.68112.0167631.66941.9983831.66341.9889 42.13192.7764241.71092.0639441.68022.0154641.66901.9977841.66321.9886 52.01502.5706251.70812.0596451.67942.0141651.66861.9971851.66301.9883 61.94322.4469261.70562.0555461.67872.0129661.66831.9966861.66281.9879 71.89462.3646271.70332.0518471.67792.0117671.66791.9960871.66261.9876 81.85952.3060281.70112.0484481.67722.0106681.66761.9955881.66231.9873 91.83312.2621291.69912.0452491.67662.0096691.66731.9950891.66221.9870 101.81242.2282301.69732.0423501.67592.0086701.66691.9944901.66201.9867 111.79592.2010311.69552.0395511.67532.0076711.66661.9939911.66181.9864 121.78232.1788321.69392.0369521.67472.0066721.66631.9935921.66161.9861 131.77092.1604331.69242.0345531.67412.0057731.66601.9930931.66141.9858 141.76132.1448341.69092.0322541.67362.0049741.66571.9925941.66121.9855 151.75302.1314351.68962.0301551.67302.0041751.66541.9921951.66101.9852 161.74592.1199361.68832.0281561.67252.0032761.66521.9917961.66091.9850 171.73962.1098371.68712.0262571.67202.0025771.66491.9913971.66071.9847 181.73412.1009381.68592.0244581.67152.0017781.66461.9909981.66061.9845 191.72912.0930391.68492.0227591.67112.0010791.66441.9904991.66041.9842 201.72472.0860401.68392.0211601.67062.0003801.66411.99011001.66021.9840

8. Stating the Conclusion 6. State the conclusion. We reject the null hypothesis that the bulbs were drawn from a population in which the average life is 1000 hrs. The difference between our sample mean (867) and the mean of the population (1000) is SO different that it is unlikely that our sample could have been drawn from a population with an average life of 1000 hours.

9. SPSS Results Computers print p values rather than critical values. If p (Sig.) is less than.05, it’s significant.

10. t-tests with Two Samples Independent Samples t-test Dependent Samples t-test

11. Independent Samples t-test Used when we have two independent samples, e.g., treatment and control groups. Formula is: Terms in the numerator are the sample means. Term in the denominator is the standard error of the difference between means.

12. Independent samples t-test The formula for the standard error of the difference in means: Suppose we study the effect of caffeine on a motor test where the task is to keep a the mouse centered on a moving dot. Everyone gets a drink; half get caffeine, half get placebo; nobody knows who got what.

13. Independent Sample Data (Data are time off task) Experimental (Caff)Control (No Caffeine) 1221 1418 1014 820 1611 519 38 912 1113 15 N 1 =9, M 1 =9.778, SD 1 =4.1164N 2 =10, M 2 =15.1, SD 2 =4.2805

14. Independent Sample Steps(1) 1. Set alpha. Alpha =.05 2. State Hypotheses. Null is H 0 :  1 =  2. Alternative is H 1 :  1   2.

15. Independent Sample Steps(2) 3. Calculate test statistic:

16. Independent Sample Steps (3) 4. Determine the critical value. Alpha is.05, 2 tails, and df = N1+N2-2 or 10+9- 2 = 17. The value is 2.11. 5. State decision rule. If |-2.758| > 2.11, then reject the null. 6. Conclusion: Reject the null. the population means are different. Caffeine has an effect on the motor pursuit task.

17. α(1 tail)0.050.0250.050.0250.050.0250.050.0250.050.025 α(2 tail)0.100.0500.100.0500.100.0500.100.0500.100.050 df 16.313812.707211.72072.0796411.68292.0196611.67021.9996811.66391.9897 22.92004.3026221.71722.0739421.68202.0181621.66981.9990821.66361.9893 32.35343.1824231.71392.0686431.68112.0167631.66941.9983831.66341.9889 42.13192.7764241.71092.0639441.68022.0154641.66901.9977841.66321.9886 52.01502.5706251.70812.0596451.67942.0141651.66861.9971851.66301.9883 61.94322.4469261.70562.0555461.67872.0129661.66831.9966861.66281.9879 71.89462.3646271.70332.0518471.67792.0117671.66791.9960871.66261.9876 81.85952.3060281.70112.0484481.67722.0106681.66761.9955881.66231.9873 91.83312.2621291.69912.0452491.67662.0096691.66731.9950891.66221.9870 101.81242.2282301.69732.0423501.67592.0086701.66691.9944901.66201.9867 111.79592.2010311.69552.0395511.67532.0076711.66661.9939911.66181.9864 121.78232.1788321.69392.0369521.67472.0066721.66631.9935921.66161.9861 131.77092.1604331.69242.0345531.67412.0057731.66601.9930931.66141.9858 141.76132.1448341.69092.0322541.67362.0049741.66571.9925941.66121.9855 151.75302.1314351.68962.0301551.67302.0041751.66541.9921951.66101.9852 161.74592.1199361.68832.0281561.67252.0032761.66521.9917961.66091.9850 171.73962.1098371.68712.0262571.67202.0025771.66491.9913971.66071.9847 181.73412.1009381.68592.0244581.67152.0017781.66461.9909981.66061.9845 191.72912.0930391.68492.0227591.67112.0010791.66441.9904991.66041.9842 201.72472.0860401.68392.0211601.67062.0003801.66411.99011001.66021.9840

18. Using SPSS

19. Independent Samples Exercise ExperimentalControl 1220 1418 1014 820 16 Work this problem by hand and with SPSS. You will have to enter the data into SPSS.

20. SPSS Results

21. Dependent Samples t-tests

22. Dependent Samples t-test Used when we have dependent samples – matched, paired or tied somehow Repeated measures Brother & sister, husband & wife Left hand, right hand, etc. Useful to control individual differences. Can result in more powerful test than independent samples t-test.

23. Dependent Samples t Formulas: t is the difference in means over a standard error. The standard error is found by finding the difference between each pair of observations. The standard deviation of these difference is SD D. Divide SD D by sqrt(number of pairs) to get SE diff.

24. Another way to write the formula

25. Dependent Samples t example PersonTX (time in sec) PlaceboDifference 160555 2352015 3706010 450455 560 0 M 55487 SD 13.2316.815.70

26. Dependent Samples t Example (2) 1.Set alpha =.05 2.Null hypothesis: H 0 :  1 =  2. Alternative is H 1 :  1   2. 3.Calculate the test statistic:

27. Dependent Samples t Example (3) 4. Determine the critical value of t. Alpha =.05, tails=2 df = N(pairs)-1 =5-1=4. Critical value is 2.776 5. Decision rule: is absolute value of sample value larger than critical value? 6. Conclusion. Not (quite) significant. Painfree does not have an effect.

28. Using SPSS for dependent t-test