Question

# 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

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 &mu; 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).

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)