Student's t-test
Student's distribution and its degrees of freedom, then the three tests on means: one sample against a value, two independent samples, two paired series; statistic, reading the table and conditions of use.
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Student's distribution: what is it?
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A bell-shaped distribution centred on zero, more spread out than the normal; it describes a standardised mean when the standard deviation is estimated from the sample.
Estimating the standard deviation adds uncertainty: tails are heavier, so thresholds are further out (2.131 against 1.96 for 15 degrees of freedom). With many degrees of freedom, it merges with the normal.
16 measurements have a mean of 52 and a standard deviation of 8. We test µ = 48 at the two-sided 5% level (Student threshold: 2.131). What conclusion?
- t = 2 < 2.131: do not reject H₀
- t = 2 > 1.96: reject H₀ at the 5% level
- t = 0.5: do not reject H₀ either
- t = 8: reject H₀
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t = 2 < 2.131: do not reject H₀
Standard error: 8 / √16 = 2; t = (52 − 48) / 2 = 2. The Student threshold at 15 degrees of freedom is 2.131, above 1.96: using the normal here would wrongly reject.
For one sample, the t statistic is (observed mean − tested value) divided by …, with … degrees of freedom.
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For one sample, the t statistic is (observed mean − tested value) divided by s / √n, with n − 1 degrees of freedom.
s is the standard deviation estimated from the sample, dividing by n − 1. The denominator is the standard error of the mean.
Student's t-test: what is it?
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A test on one or two means whose statistic, a gap divided by the estimated standard error, follows Student's distribution.
It assumes roughly normal data or a large enough sample. Three variants: one sample against a value, two independent samples, two paired series.