The independent-means t-test: Answers the question: is there a "real" difference between the two conditions in my experiment? Or is the difference due to chance? Previous lecture: (a) Dependent-means t-test: ( Matched-pairs" or "one-sample" t-test). Same subjects do both experimental conditions e.g., two conditions A and B: half subjects do A then B; rest do B then A. (Randomly allocated to one order or the other).
This lecture: (b) Independent-means t-test: ( Two-sample" t-test). Different subjects do each experimental condition. e.g., two conditions A and B: half subjects do A; rest do B. (Randomly allocated to A or B). The t-test formula: Same logic as z-score - Observed difference between sample means - hypothesised difference Estimate of how much successive sample means differ spontaneously t ( X ) ( µ X µ ) ( X X ) + ( ) ( NX ) + ( N ) hypothesis ed N X + N
Independent Means t-test, step-by-step: Effects of Prozac on driving ability. Two separate groups of subjects. Group X: ten subjects take a driving test after taking Prozac. Group : ten subjects take a driving test without any drugs. Group X: Group : X X Subject 38 444 Subject 47 09 Subject 35 5 Subject 45 05........ Subject 0 4 68 Subject 0 48 304 t ( X ) ( µ X µ ) ( X X ) + ( ) ( NX ) + ( N ) hypothesis ed N X + N Step : N X the number of subjects in condition X. Here, N X 0. N the number of subjects in condition. Here, N 0. Step : Add together all the X scores, to get the sum of X: ΣX 389 3
Step 3: Divide the result of Step (i.e., ΣX) by the number of subjects who did condition X, to get the mean of the X scores: X X 389 NX 0 38. 9 Step 4: Add together all the scores, to get the sum of : Σ 44 Step 5: Divide Σ by the number of subjects who did condition, to get the mean of the scores: 44 N 0 44. 4
Step 6: Square each X score, and then add together all these squared scores, to get ΣX 545 Step 7: Do the same with the scores, to get Σ 960 Step 8: Add together all the X scores (which we have already done in step above) and then square this sum: (ΣX) 389 Step 9: Do the same with the scores: (Σ) 44 Step 0: Work out ( X X) ( X X ) X ( X ) NX 545 389 0 3 Step : Likewise, ( ) ( ) N 9605 44 0 74 5
Step : (µx - µ) hyp. is usually zero. Step 3: t ( 38.9 44.) ( 0) 3 + 74 * + 9 + 9 0 0 87 8 5.3 * ( 0.+ 0.) t 5.3 0.389 * 0. 5.3.078 5.3.44 t - 3.68 with (n X -) + (n - ) degrees of freedom. critical values of t (two-tailed): Compare obtained value of t to the critical value of t for 8 d.f., found in a table of critical t-values (same table as for repeated-measures t- test). The critical value of t (for a twotailed test with a p level of 0.05 ) is.0. Our obtained t is bigger than this critical value. We conclude that there is a statistically significant difference between the two groups: driving ability is affected by Prozac ( t (8) - 3.68, p<0.05). 6
Our obtained value of t is even less likely to occur by chance than the critical value in the table: -3.68 -.0 +.0 as t increases, it becomes less likely to occur by chance Requirements for performing a t-test: Both t-tests are parametric tests; they assume the data possess certain characteristics or "parameters": (a) The frequency of obtaining scores in each group or condition should be roughly normally distributed. (b) The data should consist of measurements on an interval or ratio scale. (c) The two groups or conditions should show "homogeneity of variance". i.e., the spread of scores within each group should be similar. 7
t-tests are "robust" with respect to violations of these assumptions, as long as the samples are not too small and there are equal numbers of subjects in both groups (in the case of the independent-means t-test). If the data do not satisfy the requirements for a t-test, use a non-parametric test - the Mann-Whitney test instead of an independent-means t-test; the Wilcoxon test instead of a repeated-measures t-test. Using SPSS for independent-means t-tests: Do different ways of parking a police car affect people s ability to detect that it is parked? Two groups see a video taken from a driver's viewpoint and press a button as soon as they see a parked police car. Only difference between the groups is the car s orientation. Group A (n 0): see car parked at an angle. Group B (n 7): see car parked straight. I.V.: orientation of parking (angle or straight). D.V.: reaction-time (msec) to detect parked car. 8
Results of analysis using SPSS: (Analyze > compare means > Independent Samples t-test). RT group N Mean Std. Deviation Std. Error Mean angle 0 4.43860.76454.7096 straight 7 5.506.94058.8 Levene's Test for Equality of Variances F Sig. t df Sig. (- tailed) t-test for Equality of Means Mean Difference Std. Error Difference 95% Confidence Interval of the Difference: Lower: Upper: Equal.03.88 -.449 35.09 -.68646.805 -.554 -.75 variances assumed Equal variances not assumed -.408 30.89.0 -.68646.8507 -.6800 -.049 9