STAT 400 Chap.10 Hypothesis Tests for Means, Proportions and Variances
Hypothesis Tests for Means, Proportions and Variances
A test is a structured argument
The course introduces hypothesis testing through the comparison with a criminal trial. The null hypothesis plays the presumption of innocence and is kept unless the evidence against it is strong enough. Rejecting it is a guilty verdict; failing to reject is not guilty, which is different from innocent.
The significance level α is the probability of rejecting a true null hypothesis, a Type I error, and it is chosen before the data are examined. A Type II error is failing to reject a false null hypothesis.
Three kinds of evidence
The same decision can be reached three ways.
Compare the test statistic with a critical value; compare the p-value with α; or, for a two-sided test, check whether the null value lies inside the matching confidence interval. The p-value is the probability, computed assuming the null hypothesis, of a result at least as extreme as the one observed.
It is not the probability that the null hypothesis is true.
One-sample tests
For a mean, the statistic is z when σ is known and t with n − 1 degrees of freedom otherwise. For a proportion, the standard error uses the null value p0, because the test asks how surprising the sample would be if the null hypothesis held; this is a deliberate difference from the confidence interval.
For a variance, the statistic (n − 1)S²/σ0² is compared with a chi-square distribution. The alternative hypothesis, chosen from the research question, decides whether the rejection region is in the upper tail, the lower tail or split across both.
Two-sample tests
Comparing two means uses the difference of sample means divided by its standard error. With known variances the reference is normal.
With unknown variances the course covers both the pooled version, which assumes equal population variances and uses n + m − 2 degrees of freedom, and the unpooled version.
Comparing two proportions uses a pooled sample proportion under the null hypothesis of equal proportions.
Writing the conclusion
A complete answer states both hypotheses with parameter symbols, the significance level, the statistic and its reference distribution, the decision, and a final sentence in the context of the problem.
Statistical significance describes the strength of evidence, not the size or importance of an effect, so reporting the estimate alongside the decision is good practice.
What this chapter covers
- 01
Hypotheses, errors and significance level
- 02
Critical values, p-values and interval evidence
- 03
Tests for one mean
- 04
Tests for one proportion
- 05
Test for one variance
- 06
Two-sample tests for means and proportions
Worked example · free
A two-sided z test with its p-value
- 2Standard error = 4.2/√36 = 0.70, so z = (31.4 − 30)/0.70 = 2.00.
- 2Two-sided p-value = 2(1 − Φ(2.00)) = 2(0.0228) = 0.0455.
- 2Since 0.0455 < 0.05, reject H0: the process mean appears to differ from 30 g, on the high side.
Key terms
- Null Hypothesis
- The statement about a parameter that is assumed true unless the data provide strong evidence against it.
- Type I Error
- Rejecting a null hypothesis that is actually true; its probability is the significance level α.
- Type II Error
- Failing to reject a null hypothesis that is actually false; its probability is written β.
- P-Value
- The probability, computed under the null hypothesis, of a test statistic at least as extreme as the one observed.
- Pooled Variance
- A weighted average of two sample variances used in the two-sample t test when equal population variances are assumed.
Hypothesis Tests for Means, Proportions and Variances FAQ
Does failing to reject the null hypothesis prove it is true?
No. It means the data are compatible with the null hypothesis at the chosen level. Like a not guilty verdict, it reflects insufficient evidence rather than proof of innocence.
Why does the proportion test use p0 in the standard error?
The test computes how unusual the sample would be if the null hypothesis were true, so every probability, including the standard error, is evaluated at the null value p0.
When should I pool the variances in a two-sample t test?
Pool only when it is reasonable to assume the two populations share a variance, for instance when the sample standard deviations are similar. Otherwise use the unpooled statistic.
Exam move
Practise writing complete test answers in the same order every time: hypotheses, α, statistic with its distribution, decision, and a context sentence. Then redo one test all three ways, critical value, p-value and interval, and confirm they agree.
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