AI HL · 13.4
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The paired-sample t-test
13.4.13
Screenshot and video to come
Each dot is one value. Drag the orange marker under a sample to move it, and the teal point at the end of its bracket to change how spread out it is. The curve below is the t distribution: the orange area is the p-value.
The test assumes the population (for a paired test, the differences) is normally distributed; two samples also need equal variances. The markers move with the mouse, a finger, or the arrow keys once selected (Tab).
Drag a marker to start.
A coach times six sprinters over 100 m before and after a training programme. Times in seconds. Before: 11.8, 12.4, 11.5, 12.9, 12.1, 12.6. After: 11.6, 12.3, 11.5, 12.5, 12.0, 12.3. She wants to test whether the programme reduces the mean time. Form the differences, state the hypotheses and state one advantage of a paired test over an unpaired test.
Have a go first. Then open the working one line at a time.
The same six sprinters are timed twice
Each 'before' time belongs with one 'after' time. These are paired data, not two independent samples.
d = after − before: −0.2, −0.1, 0, −0.4, −0.1, −0.3
Form one new list. Choose the order once and keep it for every pair.
μ_d is the population mean change in time. H0: μ_d = 0, H1: μ_d < 0
No change under H0. The programme reduces the time, so d is negative: a lower tail.
A paired test removes the differences between sprinters, so it detects only the change caused by the programme.
Some sprinters are faster than others. Pairing stops that spread from hiding the effect.
A physiotherapist tests whether a stretching routine reduces the time, in minutes, that 8 runners take to run 1 km. Before: 5.42, 5.15, 6.01, 5.78, 5.33, 5.9, 5.6, 5.25. After: 5.31, 5.12, 5.85, 5.8, 5.21, 5.72, 5.49, 5.2. Assume that the differences are normally distributed. Test, at the 5% significance level, whether the routine reduces the mean time.
Have a go first. Then open the working one line at a time.
d = after − before: −0.11, −0.03, −0.16, 0.02, −0.12, −0.18, −0.11, −0.05
Enter these in a list.
μ_d is the population mean change in time. H0: μ_d = 0, H1: μ_d < 0
The routine reduces the time, so the mean difference is below 0.
t-test on the list d, μ0 = 0, μ < μ0: t = −3.87, p = 0.00306
A paired test is a one-sample t-test on the differences.
0.00306 < 0.05, so reject H0. There is sufficient evidence, at the 5% significance level, that the stretching routine reduces the mean time to run 1 km.
Answer in context.
A shop tests new packaging in 7 stores. The weekly sales, in units, in each store were recorded before and after the change. Before: 212, 185, 240, 198, 225, 170, 205. After: 221, 190, 243, 209, 224, 181, 214. Assume that the differences are normally distributed. Test, at the 5% significance level, whether the new packaging increases mean weekly sales.
Have a go first. Then open the working one line at a time.
d = after − before: 9, 5, 3, 11, −1, 11, 9
A rise in sales gives a positive d.
μ_d is the population mean change in weekly sales. H0: μ_d = 0, H1: μ_d > 0
An increase is an upper tail.
t-test on the list d, μ0 = 0, μ > μ0: t = 3.92, p = 0.00392
Pick the tail that matches H1.
0.00392 < 0.05, so reject H0. There is sufficient evidence, at the 5% significance level, that the new packaging increases mean weekly sales.
Answer in context.
In the shop example above, state the distributional assumption needed for the paired t-test to be valid.
Have a go first. Then open the working one line at a time.
A one-sample t-test on d
The differences are the single sample.
The population of differences is normally distributed.
It is the differences that must be normal. The before and after values need not be.