AI HL · 13.4
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Two-sample tests for means: z and pooled t
13.4.12
Screenshot and video to come
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 dietician claims that adults who eat breakfast have a lower mean resting heart rate than adults who skip it. Random samples of each group are taken. State the null and alternative hypotheses, defining each mean.
Have a go first. Then open the working one line at a time.
μ1 = population mean resting heart rate of adults who eat breakfast; μ2 = population mean resting heart rate of adults who skip it
Group 1 is the group named first in the claim. Always say in words what each mean is.
H0: μ1 = μ2
H0 says there is no difference between the two population means.
H1: μ1 < μ2
'Lower' for group 1 is one-tailed. 'Different' would be μ1 ≠ μ2.
Two machines cut portions of cheese. The masses, in grams, are normally distributed with known standard deviations of 3.1 g for machine 1 and 2.6 g for machine 2. A random sample of 15 portions from machine 1 has mean mass 48.2 g. A random sample of 20 portions from machine 2 has mean mass 49.9 g. Test, at the 5% significance level, whether machine 1 cuts lighter portions on average.
Have a go first. Then open the working one line at a time.
μ1, μ2: population mean mass of a portion from machine 1, machine 2. H0: μ1 = μ2, H1: μ1 < μ2
'Lighter' for machine 1 gives a lower tail.
Two-sample z-test, because both σ are known
The standard deviations are population values, not estimates from the samples.
Two-sample z-test from statistics: σ1 = 3.1, n1 = 15, x̄1 = 48.2, σ2 = 2.6, n2 = 20, x̄2 = 49.9, μ1 < μ2: z = −1.72, p = 0.0429
Enter the σ values as given. The test does not pool anything.
0.0429 < 0.05, so reject H0. There is sufficient evidence, at the 5% significance level, that machine 1 cuts lighter portions on average.
Compare p with 0.05 and answer in context.
Two driving routes are timed, in minutes. The journey times on each route are normally distributed with standard deviation 1.5 minutes. Route A: 24.1, 26.3, 25.2, 27.0, 25.8, 26.4. Route B: 23.5, 24.8, 25.1, 23.9, 25.6, 24.2, 24.7. Test, at the 10% significance level, whether route A takes longer on average. Find the p-value.
Have a go first. Then open the working one line at a time.
μ1, μ2: population mean time on route A, route B. H0: μ1 = μ2, H1: μ1 > μ2
'Takes longer' is an upper tail.
Two-sample z-test from lists: list 1 = route A, list 2 = route B, σ1 = 1.5, σ2 = 1.5, μ1 > μ2: z = 1.51, p = 0.0660
With lists you enter σ yourself. The calculator finds the sample means.
0.0660 < 0.1, so reject H0. There is sufficient evidence, at the 10% significance level, that route A takes longer on average.
At the 10% level the evidence is enough. At 5% it would not be.
A biologist compares the heights of two varieties of plant. A random sample of 11 plants of variety 1 has mean height 23.4 cm and standard deviation 2.6 cm. A random sample of 13 plants of variety 2 has mean height 25.9 cm and standard deviation 3.0 cm. Heights are normally distributed with equal variances. Find the p-value for a test of whether the mean heights differ.
Have a go first. Then open the working one line at a time.
H0: μ1 = μ2, H1: μ1 ≠ μ2
'Differ' means two tails.
Two-sample t-test, pooled
The population σ are unknown and assumed equal. The 2.6 and 3.0 are sample standard deviations, so this is a t-test.
Two-sample t-test from statistics, pooled: x̄1 = 23.4, s1 = 2.6, n1 = 11, x̄2 = 25.9, s2 = 3.0, n2 = 13, μ1 ≠ μ2: t = −2.16, p = 0.0419
The p-value already covers both tails.
The lifetimes, in hours, of a sample of 7 batteries of brand A and 8 batteries of brand B are shown. Brand A: 312, 298, 325, 301, 310, 290, 305. Brand B: 330, 321, 318, 335, 309, 326, 340, 322. Lifetimes are normally distributed with equal variances. Test, at the 5% significance level, whether brand A batteries last less long on average.
Have a go first. Then open the working one line at a time.
μ1, μ2: population mean lifetime of brand A, brand B. H0: μ1 = μ2, H1: μ1 < μ2
'Last less long' is a lower tail.
Two-sample t-test from lists, pooled, μ1 < μ2: t = −3.54, p = 0.00182
Put brand A in list 1 and brand B in list 2, so that the tail matches H1.
0.00182 < 0.05, so reject H0. There is sufficient evidence, at the 5% significance level, that the mean lifetime of brand A batteries is less than that of brand B.
Name both brands in the conclusion. State the assumption of normality and equal variances if asked.