The relationship between confidence intervals and hypothesis testing is a fundamental concept in statistics. Specifically, when you perform hypothesis testing, the decision you make can often be linked to confidence intervals.
In the case where you have a null hypothesis \( H_{0}: \mu = \mu_{0} \) and an alternative hypothesis \( H_{1}: \mu \neq \mu_{0} \), a 99% confidence interval gives you a range of values that you can be 99% confident contains the true population mean \(\mu\).
If you compute a 99% confidence interval and find that it does not contain the value \(\mu_{0}\) (the value specified under the null hypothesis), this suggests that the observed data is sufficiently inconsistent with the null hypothesis, therefore you would reject \( H_{0} \).
Thus, if a 99% confidence interval does not contain \(\mu_{0}\), we reject at the 1% level (the significance level \(\alpha\) for a two-tailed test is \(1 - 0.99 = 0.01\)).
In summary, the statement would read as follows:
If a 99% confidence interval does not contain \(\mu_{0}\), we reject at the 1% level.
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