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 critical region approach. There is is one more approach we can use: the confidence interval approach. For the \(t\)-test, all three approaches will lead us to the same conclusion.
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 \(\mu = \mu_0\).