91影视

The null hypothesis is \({H_0}:{\mu _1} - {\mu _2} = 0\), \({\mu _2}\) with representing the true average of alloy and \({\mu _1}\)being the true average of steel beam. The type II error \(\beta \) for \({\mu _1} - {\mu _2} = {\Delta ^'}\) vary depending on the alternative hypothesis. The alternative hypothesis is \({H_a}:{\mu _1} - {\mu _2} > 0\) and \({\Delta ^'} = 0.4, \), which indicates that the type II error is present

\(\beta \left( {{\Delta ^'}} \right) = \Phi \left( {{z_\alpha } - \frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }} \right)\)

Where

\(\sigma = \sqrt {\frac{{\sigma _1^2}}{m} + \frac{{\sigma _2^2}}{n}} \)

\({\sigma _1} = {\sigma _2} = 0.05\)and \(m = n\). The significance level is \(\alpha = 0.01,{z_\alpha } = {z_{0.01}} = 2.33\). Using the fact

\(\Phi (1.645) = 0.05\)

The following holds true, as determined by the table in the book's appendix.

\(\begin{array}{l}\beta \left( {{\Delta ^'}} \right) = \Phi \left( {{z_\alpha } - \frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }} \right)\\0.05 = \Phi \left( {{z_\alpha } - \frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }} \right) and 0.05 = \Phi (1.645)\\\Phi (1.645) = \Phi \left( {{z_\alpha } - \frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }} \right)\end{array}\)

The following is true because \(\Phi \) is a cdf of standard normal distribution.

\(\begin{array}{l}{z_\alpha } - \frac{{{\Delta ^'} - {\Delta _0}}}{\sigma } = 1.645\\2.33 - \frac{{0.04 - 0}}{\sigma } = 1.28\\{\sigma ^2} = \frac{{0.0{4^2}}}{{{{(1.645 + 2.33)}^2}}}\end{array}\)

remember that

\(\sigma = \sqrt {\frac{{\sigma _1^2}}{m} + \frac{{\sigma _2^2}}{n}} ,\)

and

\({\sigma ^2} = \frac{{\sigma _1^2}}{m} + \frac{{\sigma _2^2}}{n} = \frac{{0.0{5^2}}}{n} + \frac{{0.0{5^2}}}{n} = \frac{{0.0025 + 0.0025}}{n}\)

by substituting this, the following is true

\(\begin{array}{l}\frac{{0.0025 + 0.0025}}{n} = \frac{{0.0016}}{{{{(1.645 + 2.33)}^2}}}\\n = \frac{{(0.0025 + 0.0025) \times {{(2.33 + 1.645)}^2}}}{{0.0016}}\\n = 49.38\end{array}\)

However, because \(n\) must be an integer number (sample size), the required value of \(n\) is

\(n = 50\)

\(n = 50\)