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The T-Test in Research

The T-TEST in SPSS

• Alternative models
• Independent samples
• Paired Samples (Matched Measures) -- for samples of matched cases
• Historically, voting turnout has been lower in southern states than in states outside the south.
• Did this pattern continue in the 2000 presidential election?
• Did the eleven states in the "old confederacy" reflect a lower rate of voting than the other states?
• Testing this hypothesis with the "nustates" data set:
• H1 : States outside the south had a higher voting turnout in 2000 than states in the "old south."
• H0 : States in the old south had a turnout that was higher or equal to turnout in non-southern states.
• A test of H1 is a directional hypothesis, and the data are in the direction that might disprove the null hypothesis, H0.
• The T-TEST procedure in SPSS 10 produces this analysis of the difference:
t-tests for Independent Samples

 Group Statistics Former confederate state, 1=yes, 0=no N Mean Std. Deviation Std. Error Mean % turnout in 2000 election 0 40 55.155 6.977 1.103 1 11 48.755 3.289 0.992 Mean Difference = 6.395

 Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means F Sig. t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference % turnout in 2000 election Lower Upper Equal variances assumed 5.647 0.021 2.938 49 0.005 6.4 2.179 2.022 10.779 Equal variances not assumed 4.315 35.942 0 6.4 1.483 3.392 9.409

• We see from the T-Test of voting turnout that the difference is significant at .01 for a two-tailed test, but what about a one-tailed test?
• The answer is that one should halve the 2-tailed probability to get a one-tailed probability level
• Thus, the difference in means for the above table is significant at a smaller level
• A two-tailed test assumes this model of testing for critical regions in both tails;
• But a one-tailed test assumes a model that concentrates the critical region in only one tail:
• Thus the null hypothesis can be rejected at .05 with a smaller absolute critical value of t, for example, 1.65 is less than 1.96
• Assuming a two-tailed test, the observed t-value of plus or minus 3.03 has .0015 in each tail of the distribution
• Because we are making a one-tailed test, we are interested only in the .005 associated with the critical value, 3.03
• Thus, we divide the stated probability for a two-tailed test by 2 for a one-tailed test
• The important thing to note is that a one-tailed test makes it "easier" to reject your null hypothesis by requiring a smaller critical value to reject, and, by implication, tests statistics calculated for a two-tailed test are twice as significant (i.e., the associated probabilities are divided by two and therefore 1/2 the size)
• Suppose your data are proportions rather than means?
• A proportion is simply the mean of a dichotomized variable, measured 0 for one category, and 1 for the other
• The "mean" for this variable is simply the proportion of those in category 1 (or percent, if multiplied by 100) .
• In the special case of proportions, the standard error of a sampling distributions can use simplifications in computation:
• The variance of a dichotomous variable when p = proportion in one category and q = proportion in the other is simply, variance = pq
• so the standard deviation is the square root of pq for dichotomous proportions
• the standard deviation becomes square root of pq divided by N
• Thus, one can use the T-Test to test for differences in proportions, as well as in means.

Go here for an example using the Paired Sample T-Test.