DIFFERENCE BETWEEN TWO MEANS: THE INDEPENDENT GROUPS T-TEST

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1 DIFFERENCE BETWEEN TWO MEANS: THE INDEPENDENT GROUPS T-TEST The previous unit demonstrated how to test the difference between two means calculated from dependent or correlated observations. Difference scores, D, were calculated for each subject, and then the single-sample t statistic was used to determine whether M D differed significantly from 0, the expected value if the null hypothesis is true (i.e., if the observations in the two conditions come from populations with the same mean, then D = 0). Sometimes researchers want to test that same null hypothesis for two samples that contain independent rather than correlated observations. But when observations are independent of or uncorrelated with one another, it makes no sense to calculate difference scores; difference scores would simply be differences between two randomly paired observations. Therefore, the paired-difference t-test is inappropriate, and researchers must use a different form of the t-test, one designed for testing differences between independent groups. Th null hypothesis for this test is analogous to that for the paired-difference test, and is equivalent to assuming that the scores come from two populations with the same, that is, H0: 1 = 2 (or H0: 1-2 = 0). The alternative hypothesis would be one of two directional hypotheses, Ha: 1 > 2 or Ha: 1 < 2, or a nondirectional hypothesis, Ha: 1 2 The formula for the independent groups t-test is: The numerator of the independent groups t is the difference between our sample means minus the null hypothesis value of the population difference, 0. The numerator will be the same as for the correlated t, since M D equals the difference between the sample means. So the difference between the correlated and independent groups t is in the denominator and the associated degrees of freedom. The denominator of the independent t is the standard deviation of the difference between sample means (also called the Standard Error of the Difference between means). Its value depends on a measure of the variability within the two groups and the size of the two samples. Although it would be possible to use the separate sample variances, s 1 2 and s 2 2, to calculate the independent t, a more common practice is to pool together the variability in the two groups to obtain a better estimate of the population variance. The estimate is better because more observations are used. Note that s w 2 simply adds together or pools the SSs for the two groups in the numerator (i.e., SS 1 + SS 2 ) and the degrees of freedom in the denominator (i.e., df 1 + df 2 = {n 1-1} + {n 2-1} = n 1 + n 2-1) to compute a shared variance. This variance within groups, hence s w 2 for s within 2, is used in the denominator and determines the df for the independent t. Two degrees of freedom are lost, one for each of the two sample means. To illustrate the independent t-test, we will reanalyze the results from the hypothetical study of inhibition. Recall that errors on the game Simon Says were obtained from children in Grade 1 and Grade 2. For the paired-difference t, it was assumed that these scores came from the same children, tested one year apart. It was meaningful to compute difference scores for the children and test whether M D differed significantly from 0. Now we assume that 12 independent observations were obtained from 6 children in Grade 1 and 6 different children in Grade 2 who were not related to specific children in

2 Grade 1. Difference scores are not appropriate. The results and calculations for the independent t are shown below. Errors Grade 1 Grade x 97 x 84 x x SS SS s w 2 = ( ) / ( ) = / 10 = Ho: 1 = 2 Ho: 1 > 2 (more errors expected in Grade 1) =.05, directional (one-tailed), df = = 10 t Critical = t Observed < t Critical Do NOT reject H0 The observed value for t is not large enough to fall in the rejection region; therefore, we do not reject the null hypothesis that the two groups come from populations with the same means. Our hypothesis that errors on Simon Says, supposedly a measure of inhibitory self-regulation, decrease with age is not confirmed. It is instructive to compare the independent and correlated t-tests. The independent t of is smaller than the value of the correlated t, which is Note that the numerator of both tests is ; therefore, the difference between the two tests is in the denominator. The denominator for the independent t, , is larger than the denominator for the correlated t, This occurs because computing difference scores subtracts out or removes some of the variability within the groups; however, this will only occur when the two sets of scores are correlated with one another. We return to

3 this point later in the unit on correlation. Although the independent t lost out with respect to the denominator and the observed value of t, note that it does benefit on the df side and hence with respect to the critical value of t. Specifically, the df is larger for the independent t (df = 10) than for the correlated t (df = 5), which means that the critical value of t to reject the null hypothesis was smaller for the independent t (t Critical = 1.812) than for the dependent t (t Critical = 2.015). This benefit, however, will be less marked for larger and more typical sample sizes and is generally offset by the smaller denominator for the correlated t. Indeed, researchers trying to test for modest differences between conditions (e.g., reading times for low and high frequency words) will generally try to plan for a correlated t. They might, for example, obtain reading times for low and high frequency words from the same subjects, so that difference scores can be calculated and tested for significance using the more sensitive correlated t. SUPPLEMENTARY PROBLEMS CORRELATED AND INDEPENDENT GROUPS T-TESTS Below are some studies that can be analyzed using either the Correlated or Independent Groups T-tests. Perform the appropriate analysis and state what conclusions are warranted by each study. For some of the studies, it would be instructive to do both the correlated and independent t tests and compare the results. 1. To evaluate the effectiveness of a weed killer, a farmer divided his land into 6 equal plots and applied No Weed to half of each plot and nothing to the other half. The number of weeds per square yard are presented in the following table. Do these results warrant the conclusion that No Weed effectively reduces the growth of weeds? Plot Nothing No Weed A racing team must choose a new driver. Both of the two candidates drive 5 different cars 10 laps each and the average speed was recorded for each car and each driver. The results are presented in the following table. Should one of the drivers be preferred and, if so, which one? Car Driver A B To determine whether or not vitamin C influences the severity of colds, researchers randomly divided 16 subjects into an experimental group (8 subjects who received vitamin C supplements) and a control group (8 subjects who did not receive vitamin C supplements). For the 4 subsequent months of the winter, the number of days sick with a cold were as follows. Do these results demonstrate that vitamin C supplements reduce the severity of colds? Experimental Control

4 4. Ms. Smith teaches at an inner-city school that draws mainly core-area children. Many of her students have difficulties at school starting at quite a young age. She creates a check-list of problems outside of school (e.g., police visiting home, parents being absent, physical abuse, being propositioned on the street, having no food to eat,...). The check-list is completed by 12 students at her school and by 12 students at a nearby school that draws students from a residential area. The mean number of items (out of a maximum of 25) checked by the two groups of students are shown below. What conclusions are warranted about the different non-school environments of the two groups of children? Mean SD Inner-City Residential An automobile dealer wanted to determine which of two equally priced makes of tire would last longer. Five cars were each fitted with two tires of brand A and two tires of brand B, with one of each on the front and one of each on each side of the car. The amount of remaining tread was determined after 20,000 kilometres and the results are presented below. Is either brand significantly better than the other with alpha =.01? Car Brand A Brand B To determine whether or not vitamin supplements increase milk production in cows, a farmer randomly divides 16 cows into an experimental group (8 cows that receive supplements) and a control group (8 cows that do not receive supplements). The average milk production over the subsequent 10 days was as follows for the two groups. Do these results warrant the conclusion that vitamins enhance milk production in cows? Control Experimental To determine whether males or females do better in statistics classes, an instructor compared the average grades of the two groups on a recent test. Do the results presented below indicate that either group does better than the other? Because it is a pilot study, the researcher has decided to use alpha=.10. Mean SD n Females Males A large charity organization wanted to evaluate which of two appeal letters was more effective. One hundred letters of both types were sent to each of 5 randomly selected towns and the number of contributions for each type of letter was determined. The results are presented below. Is either letter significantly better than the other with alpha =.02? /86.7b2 Town Letter A Letter B

5 9. Social psychologists hypothesized that students would obey a questionable order more quickly from someone in authority at the University than from some unidentified person. A total of 30 students were approached either by someone who identified themselves as a University official or by someone who did not identify themselves. They were ordered to go to the nearest washroom and wait there for further instructions. Mean times in seconds to obey the instruction for each group are shown below along with the standard deviations (students who did not obey were given maximum times of 30 seconds). What conclusions are warranted about the effect of being an authority? Group Mean SD N Authority Unidentified To determine the effectiveness of reinforcing study behavior on the amount of time studying, 16 students were randomly divided into an experimental group (8 subjects who reinforced themselves for studying) and a control group (8 subjects who did not reinforce themselves for studying). The number of hours spent studying over the next four weeks was as follows for the two groups. Do these results suggest that reinforcement is an effective way to increase study time? Control Experimental To compare the effectiveness of two different anti-fouling paints used to protect ship hulls, five fishing vessels were painted on one side with brand A and on the other side with brand B. The cleanness of the hull was measured after one month of service and the results are presented in the following table. Is either brand significantly better than the other with alpha =.01? Ship Brand A Brand B A measure of Family Dysfunction (higher scores indicate more dysfunction) was completed by 6 female teenage prostitutes from a local social agency and by 6 female school students of comparable age and social class. Do the results below support the hypothesis that family dysfunction is associated with sexual exploitation of young girls? Mean SD Prostitutes School To determine whether culture A is older than culture B, archeologists determined the age of 8 randomly selected artifacts from culture A and 8 randomly selected artifacts from culture B. The age of the artifacts (in centuries) is presented below for the two cultures. Do these results warrant the conclusion that culture A is older than culture B? Culture B: Culture A:

6 14. Textbook publishers wanted to know whether one brand of paper was more durable than another. They produced five books with half of the pages being brand A and half being brand B. After 6 months of heavy use in a school library, the quality of the pages was rated. The results are in the following table. Is either brand significantly better than the other with alpha =.02? Book Brand A: Brand B: A farmer randomly assigned 7 acres to be treated with fertilizer A-098 and 6 acres to be treated with C-403. The number of bushels per acre are presented in the following table. Is fertilizer A-098 more effective than fertilizer C-403? A C Forestry researchers treated spruce bud worms in 5 plots of land with a natural insecticide, PCF-43, and treated spruce bud worms in another 8 plots with a manufactured chemical agent, XXD-87. The number of worms per tree was later determined for each plot and the results appear in the following table. Is either of these agents more effective than the other? PCF XXD To determine whether students from inner city schools do less well at university than students from suburban schools, educational psychologists at UofW tested 12 randomly selected first-year students from each type of school. The means and standard deviations are shown below for average percentage grade in first year courses. Do these results warrant the conclusion that university students from inner-city schools do less well than students from suburban schools? Type Mean SD Inner-city Suburban To determine whether males or females are more influenced by persuasive messages, 4 males and 5 females each heard 20 messages that endorsed a particular position on a number of different issues. Subjects were later asked for their beliefs on the same issues. The mean number of issues with which the subjects' beliefs agreed with the messages was for the males and 8.40 for the females. The deviations from the group mean were for the males: -2.25, 2.75,.75, -1.25; and for the females: -.40, 1.60, -2.40,.6,.6. Do these results indicate that males and females differ in how much they are influenced by persuasive messages?

7 19. Different groups who interact with delinquents often have different attitudes. For example, authority figures (e.g., police, corrections officers) may have more punitive and preventive attitudes, whereas program workers (e.g., social workers, teachers, psychologists) may have more rehabilitative and therapeutic attitudes. A measure of Punitive-Supportive attitudes (high scores mean more supportive and low scores mean more punitive) was administered to 8 Authority figures and 7 Program workers. Do the results shown below confirm the hypothesized differences between the two groups? Mean SD Program Authority Health psychologists were curious whether families of patients who experience coronary problems tend to be more or less stressed than families of patients who experience neurological problems. Spouses of patients in the two groups completed a Stress Questionnaire specifically designed to measure health-related stress. Scores for the two groups appear below (higher scores indicate more stress). What conclusions are warranted about the effect of type of illness on stress? Coronary Neurological A third group of spouses whose partners were not ill completed the Stress Questionnaire described in question 20. The results appear below. Do each of the ill groups differ from the nonill group? Do the three groups differ with respect to stress levels? [Note this last question requires analysis of variance] Non-ill

8 CORRELATED AND INDEPENDENT GROUPS T-TESTS - PART ANSWERS (Let Jim know of any errors) Legend COR = Correlated Groups TTest IND = Independent Groups TTest REJ = Reject Ho DNR = Do Not Reject Ho Type Calculation t df tcrit Conclusion 1. COR / DNR 1-tail 2. COR -3.00/ ±2.776 DNR 2-tail 3. IND / ±2.145 DNR 2-tail DNR 1-tail 4. IND 6.58/ ±2.074 REJ 2-tail REJ 1-tail 5. COR / ±4.604 DNR 2-tail 6. IND / REJ 1-tail 7. IND 2.71/ DNR 2-tail 8. COR -2.00/ ±3.747 DNR 2-tail 9. IND 8.63/ REJ 1-tail 10. IND / REJ 1-tail 11. CORR -4.60/ ±4.604 DNR 2-tail 12. IND 5.91/ REJ 1-tail ±2.228 DNR 2-tail 13. IND / DNR 1-tail 14. COR -4.20/ ±3.747 REJ 2-tail 15. IND / REJ 1-tail ±2.201 REJ 2-tail 16. IND / ±2.201 DNR 2-tail 20. IND.7413/ ±2.179 DNR 2-tail 21. ANV / F= , REJ