Connection between Hypothesis Testing and Confidence Intervals

Chapter 7. Hypothesis Testing: Introduction to Hypothesis Testing (p-value Approach)

Connection between Hypothesis Testing and Confidence Intervals

Recall that a population mean $\mu$ can be estimated using a $C\%$ confidence interval $(CI)$ .

Reinterpreting the Confidence Level C

The confidence level $C$ of a confidence interval can be reinterpreted as $(1 - \alpha)\cdot 100$ for some number $\alpha$ .

For example, if $C = 95$ , then $\alpha = 0.05$ , or if $C=99$ , then $\alpha = 0.01$ .

Connecting Hypothesis Testing and Confidence Intervals

Thus a $C\%\,CI$ for $\mu$ can be reinterpreted as a $(1-\alpha)\cdot 100\%\,CI$ for $\mu$ .

This enables us to establish a direct connection between a two-sided hypothesis test for $\mu$ and a $(1-\alpha)\cdot 100\%$ confidence interval for $\mu$ :

If $\mu_0$ falls inside the $(1 - \alpha)\cdot 100\%\,CI$ , then $H_0: \mu=\mu_0$ should not be rejected at the $\alpha$ level of significance.
If $\mu_0$ falls outside of the $(1 - \alpha)\cdot 100\%\,CI$ , then $H_0: \mu=\mu_0$ should be rejected at the $\alpha$ level of significance.

A $94\%$ confidence interval for a population mean $\mu$ , computed based on a simple random sample from the population, is $(0.428,\,\, 1.866)$ .

Suppose you use the same sample to test $H_0: \mu = 0$ against $H_a: \mu \neq 0$ at the $\alpha = 0.06$ level of significance.

What would be the conclusion?

Reject $H_0$ .

Since the $94\%$ confidence interval $(0.428,\,\,1.866)$ does not contain the value $\mu_0 = 0$ , we would reject $H_0: \mu = 0$ at the $\alpha = 0.06$ level of significance.

New example