AI SL · 13.4
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One-sample t-test for a mean
13.4.5
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 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 bakery says its loaves have a mean mass of 800 g. After a change of recipe the owner thinks the mean mass has increased. The masses of 8 randomly chosen loaves, in grams, are shown. Test at the 5% significance level. (a) State the hypotheses. (b) Find the t-statistic and the p-value. (c) State the conclusion in context. (d) State an assumption needed for this test.
Have a go first. Then open the working one line at a time.
H0: μ = 800, H1: μ > 800, where μ g is the population mean mass of a loaf.
The owner thinks the mean has increased, so H1 has >. H0 gives the old value only.
From the calculator, one-sample t-test with μ0 = 800, upper tail: t = 3.30, p = 0.00655
Put the 8 masses in a list. The calculator uses the sample's own standard deviation.
0.00655 < 0.05, so reject H0.
Compare the p-value with the significance level, 0.05.
There is sufficient evidence, at the 5% significance level, that the mean mass of a loaf has increased.
The conclusion says what it means for the loaves.
The masses of loaves are normally distributed.
A t-test is only valid if the population is normal.
A battery maker says its batteries last a mean of 14.0 hours. A reviewer thinks the mean life is less. She tests 6 batteries and records the lives in hours. Test, at the 10% significance level, whether there is evidence that the mean life is less than 14.0 hours.
Have a go first. Then open the working one line at a time.
H0: μ = 14.0, H1: μ < 14.0
The reviewer thinks the mean is less, so H1 has <.
One-sample t-test, lower tail: t = −2.14, p = 0.0426
A negative t-statistic means the sample mean is below 14.0.
0.0426 < 0.1, so reject H0.
Compare with the significance level, 0.10.
There is sufficient evidence, at the 10% significance level, that the mean battery life is less than 14.0 hours.
Write the conclusion about the batteries.
A machine is set to put 500 g of sugar in each bag. A worker thinks the mean mass has changed. The masses of 9 randomly chosen bags, in grams, are shown. Test, at the 2% significance level, whether the mean mass has changed.
Have a go first. Then open the working one line at a time.
H0: μ = 500, H1: μ ≠ 500
'Changed' means up or down, so the test has two tails.
One-sample t-test, two-tailed: t = −3.43, p = 0.00892
The calculator's p-value already includes both tails.
0.00892 < 0.02, so reject H0.
Compare p with 0.02. Do not halve the significance level.
There is sufficient evidence, at the 2% significance level, that the mean mass of a bag has changed from 500 g.
Say the mean has changed. The sample mean is below 500 g, but a two-tailed test only shows a change.
The 5 km times of the members of a running club are normally distributed with a mean of 31.5 minutes. After a training programme, a random sample of 16 members has a mean time of 29.8 minutes. The estimate of the population variance from the sample is 6.25. Test, at the 5% significance level, whether the mean time has changed.
Have a go first. Then open the working one line at a time.
H0: μ = 31.5, H1: μ ≠ 31.5
The question asks if the mean has changed: two-tailed.
s = √6.25 = 2.50
The calculator needs the standard deviation, not the variance.
Summary statistics: x̄ = 29.8, s = 2.50, n = 16, μ0 = 31.5. From the calculator: t = −2.72, p = 0.0158
Choose the t-test with statistics, not a data list.
0.0158 < 0.05, so reject H0.
Compare with 0.05.
There is sufficient evidence, at the 5% significance level, that the mean 5 km time has changed.
Say what happened to the mean time.
A tablet is advertised with a mean battery life of 10 hours. A reviewer thinks the real mean is less. She tests 12 tablets and finds a sample mean of 9.4 hours. State the null and alternative hypotheses for a one-sample t-test, defining μ.
Have a go first. Then open the working one line at a time.
Let μ hours be the population mean battery life of the tablets.
Always say what μ stands for.
H0: μ = 10
H0 is the claim being tested. It uses the population mean, never the sample mean.
H1: μ < 10
The reviewer thinks the mean is less, so the test is lower-tail.