3.12 Looking ahead

In this unit we learned how to use a sample to estimate an unknown population parameter. Rather than reporting a single number, we reported a confidence interval — a range of values consistent with the observed data.

For many problems that is exactly what is needed. Opinion polls report margins of error, medical studies estimate treatment effects, and government surveys publish confidence intervals around unemployment rates, inflation and average income. In each case the question begins with the data:

Given this sample, what values of the population parameter are plausible?

Many real questions, however, begin somewhere else.

A school claims that its students attend an average of at least 28 classes. A manufacturer advertises that its batteries last 500 hours. A government announces that unemployment has fallen below 5%.

These claims are made before any data are collected.

The task is no longer to estimate an unknown quantity. It is to decide whether a sample provides enough evidence to support — or to challenge — a specific claim that someone else has put forward.

This requires turning the question around. Instead of asking

What values are consistent with the data?

we ask

Are the data consistent with a particular value?

At first sight these look like different problems. One starts with the data and searches for plausible parameter values; the other starts with a proposed value and asks whether the data agree with it.

As the next unit shows, the two are closely connected. Hypothesis tests and confidence intervals are built from the same ingredients — the sampling distribution, the standard error, and the probability model laid on top of them — but they are pointed in opposite directions.

Confidence intervals ask what values are plausible.

Hypothesis tests ask whether a particular claim is plausible.