CVEN9407 Chap.4 Hypothesis Testing, Error Types and Power
Hypothesis Testing, Error Types and Power
Every coefficient in the project is eventually tested, and this chapter builds the testing logic on simple means first. It sets out the null and alternative hypotheses, the confidence interval and test of significance routes to a decision, and the two kinds of error. A figure shows how the significance level, the sample size and the true effect size drive power.
A worked t test on spot speeds shows why one-sided and two-sided tests can disagree, and the chapter closes with p values, the 2-t rule and comparing two independent means.
In the project, almost every claim about a factor affecting crashes rests on a test.
A coefficient reported as significant, a diagnostic that rejects constant variance, a likelihood ratio test between logit models: each one is the same reasoning applied to a different statistic. Understanding size and power also explains why a small area with few crashes may fail to show effects that are real.
What this chapter covers
- 01
Null and alternative hypotheses
- 02
Confidence interval and test of significance approaches
- 03
Type I error, type II error and power
- 04
Confidence intervals for a mean and a variance
- 05
One-sided and two-sided t tests
- 06
P values and the 2-t rule
- 07
Comparing two independent means
Worked example · free
A z test on average travel time with known spread
- 1Standard error = 12 / square root of 36 = 2 seconds.
- 1z = (104 - 100) / 2 = 2.0.
- 1The two-sided critical value is 1.96; since 2.0 > 1.96, reject the null: mean delay has changed, with a p value of about 0.046.
Key terms
- Null hypothesis
- The null hypothesis is the stated parameter value that a test assumes true until the sample evidence rejects it.
- Power of a test
- The power of a test is the probability that it rejects a null hypothesis that is false.
- P value
- A p value is the lowest significance level at which the null hypothesis can be rejected with the observed data.
- Type II error
- A type II error is the failure to reject a null hypothesis that is actually false.
Hypothesis Testing, Error Types and Power FAQ
What does it mean to fail to reject the null?
Only that the sample gives no strong reason to reject it. The null is not proven, and another hypothesis might fit the data equally well. Report the estimate and its interval so a reader can see what values remain plausible.
How can I increase the power of a test?
Collect more observations, accept a higher significance level, or study a larger true effect. Raising the level from 5% to 10% increases power but doubles the chance of a false positive, so it should be decided before seeing the data.
When do I use z and when do I use t?
Use z when the population standard deviation is known or the sample is large. Use t, with n minus one degrees of freedom, when the standard deviation is estimated from a modest sample, which is the usual case in project data.
Why might a real effect fail to show up as significant in my area?
With few crashes the standard errors are large and the test has low power, so a genuine effect can produce a t ratio below the critical value. Report the estimate and its confidence interval, and say that the data are too thin to rule the effect in or out.
Is a 5% significance level compulsory?
No, but it is the convention in the lectures and in most transport research. Whatever level you choose, fix it before looking at the results, apply it consistently across every model in the report, and give p values so readers can apply a stricter or looser level themselves.
Assessment move
Write the five-step test of significance on a card and use it for every test in the course, from a single mean to a regression coefficient. Practise each problem twice, once with an interval and once with a test statistic, and check that both reach the same decision. Then explain power in a sentence without using the word probability.
For every test you report in the project, state the null in words, the statistic, the critical value or p value, and the decision in one line.
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