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Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let e be the level of confidence used to construct a confidence

Confidence intervals
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asked 2021-06-01

Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let e be the level of confidence used to construct a confidence interval from sample data. Let \(\alpha\) be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean. (A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters such as p, \(mu_{1}-\mu_{2}\), or \(p_{1}-p_{2}\), which we will study in Section 9.3, 10.2, and 10.3.) Whenever the value of k given in the null hypothesis falls outside the \(\displaystyle{c}={1}-\alpha\) confidence interval for the parameter, we reject \(H_{0}\). For example, consider a two-tailed hypothesis test with \(\alpha =0.01\ \text{and}\ H_{0}:\mu =20 H_{1}:\mu 20\) sample mean \(\displaystyle{x}¯={22}\) from a population with standard deviation \(\displaystyle\sigma={4}\).
(a) What is the value of c\(\displaystyle{c}={1}-\alpha\);. Using the methods, construct a \(\displaystyle{c}={1}-\alpha\); confidence interval for μ from the sample data. What is the value of \(\mu\); given in the null hypothesis (i.e., what is k)? Is this value in the confidence interval? Do we reject or fail to reject \(H_0\) based on this information?
(b) using methods, find the P-value for the hypothesis test. Do we reject or fail to reject \(H_0\)? Compare your result to that of part (a).

Expert Answers (1)

2021-06-02

(a)\(c=0.99=99%\)%
20.2833 to 23.7167
\(\displaystyle\mu={20}\) is not in the confidence interval
Reject the null hypothesis \(\displaystyle{H}_{{{0}}}\).
(b)Reject \(\displaystyle{H}_{{{0}}}\)
Same conclusion as in part (a)

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