91影视

The power curves of Figures 10.5 and 10.6 can be used to obtain \(\beta = P\) (type II error) for the F test in two-factor ANOVA. For fixed values of\({\alpha _1},{\alpha _2}, \ldots ,{\alpha _1}\), the quantity \({f^2} = (J/I)\Sigma \alpha _i^2/{\sigma ^2}\) is computed. Then the figure corresponding to \({v_1} = I - 1\)is entered on the horizontal axis at the value\(\phi \), the power is read on the vertical axis from the curve labeled \({\nu _2} = (I - 1)(J - 1),\;and\;\beta = 1 - \)power.

a. For the corrosion experiment described in Exercise 2, find\(\beta \;when\;{\alpha _1} = 4,{\alpha _2} = 0,{\alpha _3} = {\alpha _4} = - 2\) and \(\sigma = 4\).Repeat for \({\alpha _1} = 6,{\alpha _2} = 0,{\alpha _3} = {\alpha _4} = - 3\) and\(\sigma = 4\)

b. By symmetry, what is \(\beta \)for the test of \({H_{0B}}\;versus\;{H_{aB}}\)in Example 11.1 when \({\beta _1} = .3,{\beta _2} = {\beta _3} = {\beta _4} = - .1\) and \(\sigma = .3\)?

Step by step solution

01

Derive the equation using degree of freedom

(a) As described in the exercise, in order to find \(\beta \)first compute\(\phi \). By the formula

With degrees of freedom \({\nu _1} = I - 1 = 3{\rm{\;and\;}}{\nu _2} = (I - 1)(J - 1) = 6\) by the mentioned chart, the power at \(\alpha = 0.05\)is \({\rm{\;power\;}} \approx 0.2\)

Which means that \(\beta \)is

\(\beta = 1 - 0.2 = 0.8\)

02

Derive the given equation using degree of freedom

By the same formula we get,

With degrees of freedom \({\nu _1} = I - 1 = 3{\rm{\;and\;}}{\nu _2} = (I - 1)(J - 1) = 6\) by the mentioned chart, the power at \(\alpha = 0.05\)is \({\rm{\;power\;}} \approx 0.5\)

Which means that \(\beta \)is

\(\beta = 1 - 0.5 = 0.5\)

03

Derive the equation using degree of freedom

(b) As described in the exercise, in order to find \(\beta \)first compute\(\phi \). By the formula

With degrees of freedom \({\nu _1} = I - 1 = 3{\rm{\;and\;}}{\nu _2} = (I - 1)(J - 1) = 6\) by the mentioned chart, the power at \(\alpha = 0.05\)is \({\rm{\;power\;}} \approx 0.5\)

Which means that \(\beta \)is

\(\beta = 1 - 0.2 = 0.8\)

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!