This chapter uses the example of testing a dog breeder’s claim about the number of spots on a Dalmatian to teach hypothesis testing with a single sample. It covers differentiating Type I and Type II errors, describing hypothesis testing in general and in practice, and conducting tests for a population mean and a population proportion.
Making a decision about a claim
The chapter frames hypothesis testing as a way to make a decision about a parameter rather than simply estimate it, illustrated with a car dealer’s advertised claim or a dog breeder’s claim about spot counts, teaching students to describe hypothesis testing in general and in practice as a structured way of evaluating such claims against sample evidence.
Type I and Type II errors
Students learn to differentiate between a Type I error, incorrectly rejecting a true claim, and a Type II error, failing to reject a false claim, a distinction the chapter treats as central to understanding what a hypothesis test’s conclusion does and does not guarantee.
Testing means and proportions
The chapter covers conducting and interpreting hypothesis tests for a single population mean, both when the population standard deviation is known and when it is unknown, as well as for a single population proportion, giving students the complete one-sample toolkit before extending it to two samples in the next chapter.
This chapter situates hypothesis testing within the scientific method, using the economic theory of consumer choice and the demand curve as an example of how a theory generates a testable prediction. It covers null and alternative hypotheses, Type I and Type II errors, and the distributions used depending on what is known about the population.
Hypothesis testing as part of the scientific method
The chapter frames statistical hypothesis testing as the formal mechanism behind the scientific method, where a theory or model built on stated assumptions leads to predictions, or hypotheses, that can be tested; it illustrates this with microeconomic consumer choice theory, whose assumptions predict the negative-sloped demand curve, a prediction statistics is used to test rather than simply assert.
Null and alternative hypotheses, Type I and Type II errors
Every hypothesis test begins by stating two competing hypotheses, the null and the alternative, and the chapter explains the two ways a test can go wrong: a Type I error, rejecting a true null hypothesis, and a Type II error, failing to reject a false one, along with how the outcomes of a test relate to these error types.
Choosing the right distribution for the test
Which distribution underlies a one-sample hypothesis test depends on what is known about the population, and the chapter covers testing a single population mean when the standard deviation is known, when it is unknown, and testing a single population proportion, connecting each case back to the sampling distributions developed earlier in the course.
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.
