Maths › Statistics › Hypothesis testing: correlation and the normal
Hypothesis testing: correlation and the normal
Two more claims meet their data: that a correlation seen in a sample is a fluke, and that a normal population still has the mean it used to. Both tests run on last lesson's logic; only the distribution under the null changes.
Builds on Hypothesis testing with the binomial and Correlation and regression.
IN THIS TOPIC
- Test H₀: ρ = 0 by comparing a sample r with a given critical value.
- Use the fact that the mean of n observations of N(μ, σ²) is N(μ, σ²/n).
- Carry out a hypothesis test for the mean of a normal distribution with known σ, concluding in context.
WHAT YOU PROBABLY THINK
The sample mean of n readings has the same standard deviation as a single reading.
Is a correlation real?
A sample scatter can show correlation even when the population has none, so the honest question is whether the sample's r is too large to be luck. The hypotheses are about ρ, the population correlation: H₀: ρ = 0 against H₁: ρ > 0, ρ < 0 or ρ ≠ 0.
The exam gives the critical value for the sample size and level. The test is a single comparison: if the magnitude of the sample r beats it, reject H₀. With r = 0.62 against a one-tailed critical value of 0.55: reject, and conclude there is significant evidence of positive correlation between the named variables.
The sample mean's distribution
Averages wobble less than individuals: the mean of n independent observations of N(μ, σ²) is itself normal, same centre, spread divided by √n. That standard deviation σ/√n is the quantity every normal-mean test standardises against.
Testing a normal mean
WORKED EXAMPLE
Has the machine drifted?
Bags from a machine have masses N(μ, 4²) grams. The machine is set to μ = 30. A sample of 25 bags has mean 31.6 g. Test at the 5% level whether the mean has increased.
H₀: μ = 30, H₁: μ > 30. Under H₀ the sample mean is N(30, 4²/25), standard deviation 4/5 = 0.8.
z = (31.6 − 30)/0.8 = 2.0, and P(Z ≥ 2.0) = 0.0228.
0.0228 < 0.05: reject H₀; significant evidence at the 5% level that the mean mass has increased.
The check: 31.6 is two standard errors above 30, and two-sigma events are rare enough to notice.
Two-tailed versions split the level as before, and the conclusion vocabulary is identical to the binomial test's. The only genuinely new content is the √n: forget it and every z-value in the question collapses.
THE EXAM BIT
- Correlation tests: hypotheses in ρ, comparison against the given critical value, conclusion naming the variables.
- Normal-mean tests: write the distribution of the sample mean with σ/√n before standardising; that line carries the method.
- One tail or two is decided by the wording: “changed” is two-tailed, “increased” or “decreased” is one.
- Conclusions in context, always: reject or not, at what level, about what quantity.
CHECK YOURSELF
Readings are N(μ, 9) and H₀ sets μ = 50. A sample of 36 has mean 49.2. Find the z-value for testing H₁: μ < 50.
Show a hint
The sample mean's standard deviation is σ/√n.
Show the answer
σ/√n = 3/6 = 0.5.
z = (49.2 − 50)/0.5 = −1.6.
Correlation: compare the sample r against the given critical value, hypotheses in ρ.
A mean of n readings lives on N(μ, σ²/n): standardise with σ/√n and test as usual.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Test H₀: ρ = 0 by comparing a sample r with a given critical value.
- Use the fact that the mean of n observations of N(μ, σ²) is N(μ, σ²/n).
- Carry out a hypothesis test for the mean of a normal distribution with known σ, concluding in context.
No animated video for this topic yet; these notes stand alone.