## 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 $$\mu = \mu_0$$.