Paired t-Test Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Five paired differences are: {2,โˆ’1,3,0,4}\{2, -1, 3, 0, 4\}. Calculate dห‰\bar{d} and sds_d, then set up the t-test statistic formula.

Solution

  1. 1
    dห‰=(2โˆ’1+3+0+4)/5=8/5=1.6\bar{d} = (2-1+3+0+4)/5 = 8/5 = 1.6
  2. 2
    Deviations: 0.4,โˆ’2.6,1.4,โˆ’1.6,2.40.4, -2.6, 1.4, -1.6, 2.4; squared: 0.16,6.76,1.96,2.56,5.760.16, 6.76, 1.96, 2.56, 5.76; sum =17.2= 17.2
  3. 3
    sd=17.2/4=4.3โ‰ˆ2.07s_d = \sqrt{17.2/4} = \sqrt{4.3} \approx 2.07
  4. 4
    t-statistic: t=dห‰sd/n=1.62.07/5=1.60.926โ‰ˆ1.73t = \frac{\bar{d}}{s_d/\sqrt{n}} = \frac{1.6}{2.07/\sqrt{5}} = \frac{1.6}{0.926} \approx 1.73

Answer

dห‰=1.6\bar{d} = 1.6, sdโ‰ˆ2.07s_d \approx 2.07, tโ‰ˆ1.73t \approx 1.73 with df=4.
The paired t-test reduces to a one-sample t-test on the differences. Calculate mean and SD of differences, then apply the one-sample t formula. With df=4 and t=1.73, we'd fail to reject Hโ‚€ at ฮฑ=0.05 (critical value โ‰ˆ2.132).

About Paired t-Test

A hypothesis test for the mean difference in a paired (matched) data design, where each subject provides two related measurements. The test analyzes the differences di=x1iโˆ’x2id_i = x_{1i} - x_{2i} as a single sample.

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