AI HL · 13.4
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Two-sample pooled t-test
13.4.6
Screenshot and video to come
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 12 swimmers who trained in the morning and 15 different swimmers who trained in the evening, and wants to compare the mean 100 m times of the two groups. Name the test to use and state its assumptions.
Have a go first. Then open the working one line at a time.
12 morning swimmers and 15 evening swimmers
Different people in each group, so the samples are independent. They need not be the same size.
Two-sample t-test, pooled
It compares two population means. 'Pooled' means the two sample variances are combined into one estimate.
Both populations are normally distributed. The two populations have equal variance.
State both. In this course the pooled test always assumes equal variance.
A teacher thinks that students who revise with flashcards (group 1) score higher on average than students who revise with notes (group 2). In the samples, the mean scores are 71.4 and 67.2. State the null and alternative hypotheses, defining μ1 and μ2.
Have a go first. Then open the working one line at a time.
μ1 is the population mean score of students who revise with flashcards. μ2 is the population mean score of students who revise with notes.
Say which group is 1 and which is 2, and say 'population mean'.
H0: μ1 = μ2
H0 says the two population means are equal. The sample means 71.4 and 67.2 are not used.
H1: μ1 > μ2
The teacher thinks flashcards give higher scores, so H1 is one-tailed with group 1 first.
A gardener grows tomato plants with fertiliser A (group 1) and fertiliser B (group 2). The heights, in cm, of random samples are summarised. She thinks that fertiliser A gives a greater mean height. Stating any assumptions, test her belief at the 5% significance level.
Have a go first. Then open the working one line at a time.
Two-sample pooled t-test. Both populations are normally distributed. The two populations have equal variance.
Two independent samples and an unknown variance, assumed equal.
H0: μ1 = μ2, H1: μ1 > μ2, where μ1 and μ2 cm are the population mean heights with fertiliser A and B.
She thinks A gives the greater mean.
Two-sample t-test with statistics, pooled: t = 2.32, p = 0.0128
Enter n, the sample mean and the estimate of s for each group. Choose pooled and the upper tail.
0.0128 < 0.05, so reject H0.
Compare p with 0.05.
There is sufficient evidence, at the 5% significance level, that the mean height of tomato plants is greater with fertiliser A than with fertiliser B.
Say which fertiliser gives the greater mean.
The journey times, in minutes, of random samples of buses on route X (group 1) and route Y (group 2) are summarised. Assume the times are normally distributed with equal variance. Test, at the 5% significance level, whether the mean journey times on the two routes are different.
Have a go first. Then open the working one line at a time.
H0: μ1 = μ2, H1: μ1 ≠ μ2
'Different' is two-tailed.
Two-sample t-test with statistics, pooled: t = 0.646, p = 0.523
Choose the two-tailed option.
0.523 > 0.05, so do not reject H0.
p is well above 0.05.
There is insufficient evidence, at the 5% significance level, that the mean journey times on the two routes are different.
The sample means differ, but the difference is small compared with the spread.
Reaction-time scores (higher is better) are recorded for 6 students after a normal night's sleep (group 1) and 7 students after a short night's sleep (group 2). The results are shown. Assume the scores are normally distributed with equal variance. Test, at the 5% significance level, whether the mean score after a normal night's sleep is less than after a short night's sleep.
Have a go first. Then open the working one line at a time.
H0: μ1 = μ2, H1: μ1 < μ2, where μ1 and μ2 are the population mean scores after a normal and a short night's sleep.
The test asks whether group 1 has the lower mean.
Two-sample t-test, lists, pooled, lower tail: t = −3.69, p = 0.00179
Put each sample in its own list. The lists can be different lengths.
0.00179 < 0.05, so reject H0.
Compare p with 0.05.
There is sufficient evidence, at the 5% significance level, that the mean score after a normal night's sleep is less than after a short night's sleep.
A negative t-statistic fits H1: the sample mean of group 1 is lower.