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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.