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 $\mu = \mu_0$.

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These notes have been prepared by Amanda J. Shaker. The copyright for the material in these notes resides with the author named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License CC BY-NC-ND