Expand Your Knowledge: Confidence Intervals and Two-Tailed

Chapter , Problem 23

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Expand Your Knowledge: Confidence Intervals and Two-Tailed Hypothesis Tests Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let c be the level of confidence used to construct a confidence interval from sample data. Let a 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 \(\alpha\) and null hypothesis \(H_0: \mu = k\), we reject \(H_0\) whenever k falls outside the \(c = 1 - \alpha\) confidence interval for \(\mu\) based on the sample data. When k falls within the \(c = 1 - \alpha\) confidence interval, we do not reject \(H_0\).

(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 Sections 9.3 and 9.5.) Whenever the value of k given in the null hypothesis falls outside the \(c = 1 - \alpha\) confidence interval for the parameter, we reject \(H_0\). For example, consider a two-tailed hypothesis test with \(\alpha = 0.01\) and

\(H_0 : \mu = 20\)                            \(H_1 : \mu\ \neq\ 20\)

A random sample of size 36 has a sample mean \(\overline{x} = 22\) from a population with standard deviation \(\sigma = 4\).

(a) What is the value of \(c = 1 - \alpha\)? Using the methods of Chapter 8, construct a \(1 - \alpha\) confidence interval for m 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 of Chapter 9, 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).