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, \(\mu1 — \mu_2,\ and\ p_{1}, - p_{2}\).
(a) Consider the hypotheses \(H_{0} : \mu_{1} — \mu_{2} = O\ and\ H_{1} : \mu_{1} — \mu_{2} \neq\) 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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d2saint0
Answered 2021-02-01 Author has 18556 answers
The level of significance, \(\alpha = 0.05\)
The null hypothesis:
\(H_{0}: \mu_{1} - \mu_{2} =0\)
The alternative hypothesis:
\(H_{0}: \mu_{1} - \mu_{2} \neq0\)
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.
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