Maths › Statistics › Hypothesis testing: correlation and the normal
Hypothesis testing: correlation and the normal
Apply the hypothesis-testing procedure to two further situations: whether a sample correlation coefficient is evidence of correlation in the population, and whether the mean of a normal population with known variance has changed. Each uses a different distribution under the null hypothesis.
Builds on Hypothesis testing with the binomial and Correlation and regression.
IN THIS TOPIC
- Test H₀: ρ = 0 against a critical value read from the table, using the right column for the number of tails.
- 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.
- Define the p-value as the probability, if H₀ is true, of a result at least as extreme as the one observed, and say what it is not.
COMMON MISCONCEPTION
The sample mean of n readings has the same standard deviation as a single reading.
The mean of n independent observations of N(μ, σ2) is itself normal, with the same mean μ and standard deviation σ/√n. Averaging 25 independent readings therefore divides the standard deviation by five, and σ/√n is the divisor when standardising in a test about a normal mean. Dividing by n instead gives incorrect z-values.
Testing a correlation coefficient
A sample can show apparent correlation when the population correlation is zero, so the test asks whether the sample value of r is too large to be attributed to sampling variation. The hypotheses are stated in terms of ρ, the population correlation coefficient: test H₀: ρ = 0 against H₁: ρ > 0, ρ < 0 or ρ ≠ 0. State the hypotheses in ρ, not in r.
The table of critical values is indexed by sample size and by one-tail probability, so select the column that matches the number of tails. A one-tailed test at 5% uses the 0.05 column; a two-tailed test at 5% uses the 0.025 column, because the significance level is divided between the tails.
The test is then a single comparison. For H₁: ρ > 0, reject H₀ when r exceeds the positive critical value. For H₁: ρ < 0, reject H₀ when r is less than the corresponding negative value. Only the two-tailed test compares |r| with the critical value. With n = 20 and r = 0.48 against H₁: ρ ≠ 0 at 5%, the table gives 0.4438, and 0.48 > 0.4438, so at the 5% significance level reject H₀ and conclude that there is sufficient evidence of correlation between the two named variables. This test assesses linear association; it does not establish causation.
The sample mean's distribution
The mean of n independent observations of N(μ, σ²) is itself normal, with the same mean μ and standard deviation σ/√n. That standard deviation is the divisor used when standardising in a test about a normal mean; dividing by n rather than √n gives incorrect z-values.
The formulae booklet prints this result in its standardised form: for a random sample of n observations from N(μ, σ²), the sample mean minus μ, divided by σ/√n, is N(0, 1). Be able to write down the distribution N(μ, σ²/n) and then standardise it.
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, so at the 5% significance level reject H₀ and conclude that there is sufficient evidence that the mean mass has increased.
Check: 31.6 is two standard deviations of the sample mean above 30, and the area beyond z = 2 is about 2.3%, consistent with the p-value obtained.
The 0.0228 is the p-value: the probability, if H₀ is true, of a result at least as extreme as the one observed, which here means a sample mean of 31.6 g or more. It is compared with the significance level, and H₀ is rejected when it is smaller. It is not the probability that H₀ is true, which no test gives; it is a probability calculated assuming H₀. In a two-tailed test a sample mean as far below 30 would count as just as extreme, so the tail probability is doubled before the comparison, or compared with half the level.
The critical-value method gives the same decision. At the 5% level, reject H₀ when z > 1.6449 for a one-tailed test, or when |z| > 1.96 for a two-tailed test. As with the binomial test, a two-tailed test divides the significance level between the tails, and the conclusion is stated in the same form.
ASSESSMENT FOCUS
- Correlation tests: hypotheses in ρ, comparison against the table's critical value, conclusion naming both variables.
- Select the table column by the number of tails: a two-tailed test at 5% uses the 0.025 column.
- For a normal-mean test, write the distribution of the sample mean, with standard deviation σ/√n, before standardising.
- The wording fixes the number of tails: “changed” is two-tailed, “increased” or “decreased” is one-tailed.
- State the conclusion in context: reject or do not reject H₀, at the stated significance level, about the named 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: hypotheses in ρ, and compare the sample r against the table value for the right number of tails.
The mean of n independent readings has distribution N(μ, σ²/n), so standardise with σ/√n.
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.
Or read them with their worked answers on the hypothesis testing: correlation and the normal questions page.
USED LATER IN
CHECK YOUR PROGRESS
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- Test H₀: ρ = 0 against a critical value read from the table, using the right column for the number of tails.
- 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.
- Define the p-value as the probability, if H₀ is true, of a result at least as extreme as the one observed, and say what it is not.
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