Chapter 6 Comparing Means The most interesting inferences about means is usually the comparison of two population means, either from an experiment or an observational study. We should always remember that the means are just one characteristic of the distributions of quantitative variables: two populations could have the same mean, but be quite different in other respects, including spread and shape. Example (Case Study 1.1.1, p. 2, in Ramsey and Schafer, The Statistical Sleuth, 2nd ed.) Do external rewards and praise stimulate people to do their best work or is intrinsic motivation more important? Psychologist Teresa Amabile (“Motivation and Creativity: Effects of Motivational Orientation on Creative Writers,” Journal of Personality and Social Psychology 48(2)(1985):393-99) conducted an experiment to compare the effects of intrinsic and extrinsic motivation on creativity. Forty-seven subjects with considerable experience in creative writing were randomly assigned to one of two treatment groups: 24 were assigned to the “intrinsic” treatment and 23 to the “extrinsic” treatment. Each group was asked to complete a questionnaire and then was asked to write a poem in the Haiku style about “laughter.” The “intrinsic” group’s questionnaire asked the subjects to rank in order of importance 7 reasons for writing, all of which related to intrinsic rewards: “You get a lot of pleasure from reading something good that you have written” and “You like to play with words” are two examples. The “extrinsic” group’s questionnaire asked the subjects to rank in order of importance 7 different reasons for writing, all of which related to extrinsic rewards: “You want your writing teachers to be favorably impressed with your work” and “You know that many of the best jobs available require good writing skills” are examples. The purpose of the questionnaires was to get the first group thinking about intrinsic rewards and the second group about extrinsic rewards. The poems the 47 subjects wrote were submitted to a panel of 12 judges who evaluated them on a 40-point scale of creativity. Judges were not told about the study’s purpose. The average ratings given by the 12 judges are shown below: Intrinsic group: 12.0 12.0 12.9 13.6 16.6 17.2 17.5 18.2 19.1 19.3 19.8 20.3 20.5 20.6 21.3 21.6 22.1 22.1 22.6 23.1 24.0 24.3 26.7 29.7 1n= 24 1y = 19.88 = 4.44 1sExtrinsic group: 5.0 5.4 6.1 10.9 11.8 12.0 12.3 14.8 15.0 16.8 17.2 17.2 17.4 17.5 18.5 18.7 18.7 19.2 19.5 20.7 21.2 22.1 24.0 2n = 23 2y= 15.74 = 5.25 2sIntrinsic ExtrinsicGroup51015202530Score2 • Do the data provide evidence that creativity scores are affected by the type of motivation induced by the questionnaires? How big might the difference be? • This is an experiment, so our inferences are to what would have happened if all 47 subjects had received the “intrinsic” treatment compared to what would have happened if all 47 subjects had received the “extrinsic” treatment. These 47 subjects are not a random sample from some larger population so, strictly speaking, we cannot generalize beyond this particular group of 47 subjects. We wish to make inferences about 21µµ−, the difference in two population means. The difference is estimated by 21yy −, the difference in the sample means. We need, therefore, to know the sampling distribution of 21yy −. First, recall the properties of the sampling distributions of 1y and 2y: E(1y) = Var(1y) = Shape if large? 1n E(2y) = Var(2y) = Shape if large? 2n Therefore, E(21yy −) = Var(21yy −) = SD(21yy −) = Shape? What assumption did you make in calculating Var(21yy−) ? Again, we must replace the unknown population standard deviations by the sample standard deviations in order to make inferences about the difference in population means. This gives us the standard error of the difference in the sample means: 22212121)(SEnsnsyy +=− • Compute SE(21yy −) for the creativity data set.3 As in the one-sample case, when we replace SD(21yy−) by SE(21yy−), we use a t distribution instead of the standard normal. That is, if we assume that the population distributions are normal, then )(SE)()(212121yyyy−−−−µµ can be closely modeled by a t distribution with degrees of freedom given by a special formula in a footnote at the bottom of page 480 of the text. A confidence interval for 21µµ− is then )(SE21*df21yytyy −±− • The special formula gives df = 43.1. Using a computer program, we find that for 99% confidence, the value of t* for 43.1 df is t* = 2.695. Use this value to compute a 99% confidence interval for 21µµ−: Computer programs like SPSS automatically compute the correct df and t*. Since the formula for the df is complicated, if you are computing a confidence interval by hand, there’s a simple rule you can use that is conservative (it will give you a little wider CI than the exact formula will). The simple rule is to take the smaller of the two sample sizes and subtract 1 and use that as the degrees of freedom. For the creativity experiment, the smaller sample size is 23, so use 22 df if doing this by hand – this gives t* = 2.819 for 99% confidence. Hypothesis testing Do the data provide evidence that the difference in the sample mean creativity scores for the two groups is not just due to the random assignment to groups? In other words, is there evidence that the mean creativity score all 47 subjects would have received under the intrinsic treatment is different from the mean score all 47 would have received under the extrinsic treatment? We want to test the hypotheses 21A210:H:Hµµµµ≠= The test statistic is )(SE2121yyyyt−−= • Compute the test statistic for the creativity experiment:4 Again, SPSS will use the df from the special formula to find the P-value. The SPSS output is given below (from Analyze…Compare Means…Independent Samples T Test): Independent Samples Test.962 .332 2.924 45 .005 4.1400 1.4161 .3313 7.94872.913 43.100 .006 4.1400 1.4213 .3100 7.9700Equal variancesassumedEqual variancesnot assumedscoreF Sig.Levene's Test forEquality of Variancest df Sig. (2-tailed)MeanDifferenceStd. ErrorDifferenceLower Upper99% ConfidenceInterval of theDifferencet-test for Equality of Means The results for the two-sample procedures we learned above are listed on the line labeled “Equal variances not
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