## 3.5 Using confidence intervals to decide whether to reject \(H_0\)

So far, we have seen how we can arrive at a decision about rejecting \(H_0\) using either the ** \(p\)-value approach** or the

**. There is is one more approach we can use: the**

*critical region approach***. For the \(t\)-test, all three approaches will lead us to the same conclusion.**

*confidence interval approach*To use the confidence interval approach, we use the following rule

- If \(\mu_0\) lies
*outside*the range of the confidence interval, reject \(H_0\) - If \(\mu_0\) lies
*within*the range of the confidence interval, do not reject \(H_0\)

Considering the cholesterol example, we have that \(\mu_0 = 5\) and our confidence interval is (5.01, 5.25). Since 5 lies outside the range of the confidence interval, we decide to reject \(H_0\).

This makes sense, because based on the confidence interval, we are saying we are 95% confident \(\mu\) is between 5.01 and 5.25. Since \(\mu_0 = 5\) is not within this range, it makes sense to say we are confident that \(\mu \neq \mu_0\), and we can therefore reject the null hypothesis that \(\mu = \mu_0\).