Presentation on theme: "Sociology 5811: T-Tests for Difference in Means"— Presentation transcript:
1Sociology 5811: T-Tests for Difference in Means Wes Longhofer, pinch-hitting for Evan Schofer
2Strategy for Mean Difference We never know true population meansSo, we never know true value of difference in meansSo, we don’t know if groups really differIf we can figure out the sampling distribution of the difference in means…We can guess the range in which it typically fallsIf it is improbable for the sampling distribution to overlap with zero, then the population means probably differAn extension of the Central Limit Theorem provides information necessary to do calculations!
3Sampling Distribution for Difference in Means The mean (Y-bar) is a variable that changes depending on the particular sample we tookSimilarly, the differences in means for two groups varies, depending on which two samples we choseThe distribution of all possible estimates of the difference in means is a sampling distribution!The “sampling distribution of differences in means”It reflects the full range of possible estimates of the difference in means.
4Mean Differences for Small Samples Sample Size: rule of thumbTotal N (of both groups) > 100 can safely be treated as “large” in most casesTotal N (of both groups) < 100 is possibly problematicTotal N (of both groups) < 60 is considered “small” in most casesIf N is small, the sampling distribution of mean difference cannot be assumed to be normalAgain, we turn to the T-distribution.
5Mean Differences for Small Samples To use T-tests for small samples, the following criteria must be met:1. Both samples are randomly drawn from normally distributed populations2. Both samples have roughly the same variance (and thus same standard deviation)To the extent that these assumptions are violated, the T-test will become less accurateCheck histogram to verify!But, in practice, T-tests are fairly robust.
6Mean Differences for Small Samples For small samples, the estimator of the Standard Error is derived from the variance of both groups (i.e. it is “pooled”)Formulas:
7Probabilities for Mean Difference A T-value may be calculated:Where (N1 + N2 – 2) refers to the number of degrees of freedomRecall, t is a “family” of distributionsLook up t-dist for “N1 + N2 -2” degrees of freedom.
8T-test for Mean Difference Back to the example: 20 boys & 20 girlsBoys: Y-bar = 72.75, s = 8.80Girls: Y-bar = 78.20, s = 9.55Let’s do a hypothesis test to see if the means differ:Use a-level of .05H0: Means are the same (mboys = mgirls)H1: Means differ (mboys ≠ mgirls).
14T-Test for Mean Difference Question: What is the critical value for a=.05, two-tailed T-test, 38 degrees of freedom (df)?Answer: Critical Value = approx. 2.03Observed T-value = 1.88Can we reject the null hypothesis (H0)?Answer: No! Not quite!We reject when t > critical value
15T-Test for Mean Difference The two-tailed test hypotheses were:Question: What hypotheses would we use for the one-tailed test?
16T-Test for Mean Difference Question: What is the critical value for a=.05, one-tailed T-test, 38 degrees of freedom (df)?Answer: Around (40 df)One-tailed test: T =1.88 > 1.684We can reject the null hypothesis!!!Moral of the story:If you have strong directional suspicions ahead of time, use a one-tailed test. It increases your chances of rejecting H0.But, it wouldn’t have made a difference at a=.01
17Another ExampleQuestion: Do the mean batting averages for American League and National League teams differ?Use a random sample of teams over timeAmerican League: Y-bar = .2677, s = .0068, N=14National League: Y-bar = .2615, s = .0063, N=16Let’s do a hypothesis test to see if the means differ:Use a-level of .05H0: Means are the same (mAmerican = mNational)H1: Means differ (mAmerican ≠ mNational)
23T-Test for Mean Difference Question: What is the critical value for a=.05, two-tailed T-test, 28 degrees of freedom (df)?Answer: Critical Value = approx. 2.05Observed T-value = 2.58Can we reject the null hypothesis (H0)?Answer: YesWe reject when t > critical valueWhat if we used an a-level of .01?Critical value=2.76
24T-Test for Mean Difference Question: What if you wanted to compare 3 or more groups, instead of just two?Example: Test scores for students in different educational tracks: honors, regular, remedialCan you use T-tests for 3+ groups?Answer: Sort of… You can do a T-test for every combination of groupse.g., honors & reg, honors & remedial, reg & remedialBut, the possibility of a Type I error proliferates… 5% for each testWith 5 groups, chance of error reaches 50%Solution: ANOVA.