Hypothesis Testing with One Sample

Summary :

Hypothesis testing uses sample data to decide between two competing claims about a population. This chapter explains how to state null and alternative hypotheses, the difference between Type I and Type II errors, and how to carry out and interpret a test for a single population mean or proportion using a p-value.

Null and Alternative Hypotheses

A hypothesis test begins with two opposing statements about a population. The null hypothesis is the claim of no effect or no difference, taken as true unless the evidence contradicts it. The alternative hypothesis is what the researcher suspects may be true instead, and it may be one-sided or two-sided depending on the question. The test uses sample data to decide whether there is enough evidence to reject the null hypothesis in favour of the alternative.

Type I and Type II Errors

Because a decision is made from a sample, it can be wrong in two ways. A Type I error occurs when a true null hypothesis is rejected, and its probability is the significance level chosen for the test. A Type II error occurs when a false null hypothesis is not rejected. Lowering the chance of one type of error tends to raise the other, so the significance level is set in advance to balance the risks according to the consequences of each mistake.

Carrying Out the Test

To test a claim about a single population mean or proportion, the appropriate distribution is chosen: the normal distribution when the population standard deviation is known, and the Student t distribution when it is estimated from the sample. A test statistic is calculated and converted into a p-value, the probability of results at least as extreme as those observed if the null hypothesis were true. If the p-value is smaller than the significance level, the null hypothesis is rejected.