 # In there a relationship between confidence intervals and two-tailed hypothesis tests? The answer is yes. Let c be the level of confidence used to cons Elleanor Mckenzie 2021-01-31 Answered
In there a relationship between confidence intervals and two-tailed hypothesis tests? The answer is yes. Let c be the level of confidence used to construct a confidence interval from sample data. Let * be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean:
For a two-tailed hypothesis test with level of significance a and null hypothesis ${H}_{0}:\mu =k$ we reject Ho whenever k falls outside the $c=1—\alpha$ confidence interval for $\mu$ based on the sample data. When A falls within the $c=1—\alpha$ confidence interval. we do reject ${H}_{0}$.
For a one-tailed hypothesis test with level of significance Ho : $\mu =k$ and null hypothesiswe reject Ho whenever A falls outsidethe $c=1—2\alpha$ confidence interval for p based on the sample data. When A falls within the $c=1—2\alpha$ confidence interval, we do not reject ${H}_{0}$.
A corresponding relationship between confidence intervals and two-tailed hypothesis tests is also valid for other parameters, such as p, .
(a) Consider the hypotheses Suppose a 95% confidence interval for ${\mu }_{1}—{\mu }_{2}$ contains only positive numbers. Should you reject the null hypothesis when $\alpha =0.05$? Why or why not?
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The level of significance, $\alpha =0.05$
The null hypothesis:
${H}_{0}:{\mu }_{1}-{\mu }_{2}=0$
The alternative hypothesis:
${H}_{0}:{\mu }_{1}-{\mu }_{2}\ne 0$
Here, from above hypothesis D = 0 and we know that for a two-tailed hypothesis test with level of significance $\alpha$, we reject ${H}_{0}$ whenever D falls outside the $c=1-\alpha$ confidence interval for $\mu$ based on the sample data. If a 95% confidence interval for ${\mu }_{1}—{\mu }_{2}$ contains only positive numbers then we have to reject ${H}_{0}$ at the level of significance $\alpha =0.05$.
Since, the confidence interval does not contain D = 0 and hence it falls outside the 95% confidence interval.