Editing Student's t-test (section)

==Worked examples==
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Let {{math|''A''<sub>1</sub>}} denote a set obtained by drawing a random sample of six measurements:

:<math>A_1=\{30.02,\ 29.99,\ 30.11,\ 29.97,\ 30.01,\ 29.99\}</math>

and let {{math|''A''<sub>2</sub>}} denote a second set obtained similarly:

:<math>A_2=\{29.89,\ 29.93,\ 29.72,\ 29.98,\ 30.02,\ 29.98\}</math>

These could be, for example, the weights of screws that were manufactured by two different machines.

We will carry out tests of the null hypothesis that the [[Arithmetic mean|mean]]s of the populations from which the two samples were taken are equal.

The difference between the two sample means, each denoted by {{math|{{overline|''X''}}<sub>''i''</sub>}}, which appears in the numerator for all the two-sample testing approaches discussed above, is

:<math>\bar{X}_1 - \bar{X}_2 = 0.095.</math>

The sample [[standard deviations]] for the two samples are approximately 0.05 and 0.11, respectively. For such small samples, a test of equality between the two population variances would not be very powerful.  Since the sample sizes are equal, the two forms of the two-sample ''t''-test will perform similarly in this example.

===Unequal variances===
If the approach for unequal variances (discussed above) is followed, the results are
:<math>\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \approx 0.04849</math>

and the degrees of freedom

:<math>\text{d.f.} \approx 7.031.</math>

The test statistic is approximately 1.959, which gives a two-tailed test ''p''-value of 0.09077.

===Equal variances ===
If the approach for equal variances (discussed above) is followed, the results are

:<math>s_p \approx 0.08399</math>

and the degrees of freedom

:<math>\text{d.f.} = 10.</math>

The test statistic is approximately equal to 1.959, which gives a two-tailed ''p''-value of 0.07857.