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A mechanical engineer wishes to compare strength properties of steel beams with similar beams made with a particular alloy. The same number of beams, n, of each type will be tested. Each beam will be set in a horizontal position with a support on each end, a force of 2500 lb will be applied at the center, and the deflection will be measured. From past experience with such beams, the engineer is willing to assume that the true standard deviation of deflection for both types of beam is 0.06 in. Because the alloy is more expensive, the engineer wishes to test at level 0.01 whether it has smaller average deflection than the steel beam. What value of n is appropriate if the desired type II error probability is 0.05 when the difference in true average deflection favors the alloy by 0.05 in.? (Round your answer up to the next whole number.)

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Answer:

50

Step-by-step explanation:

Level of significance, a = 0.01

Type II error probability, b =0.05

From the z-table:

the critical value at 1% level of significance, Zα = Z0.01 = 2.33

the critical value at 5% probability, Zβ = Z0.05 = 1.645

The critical value of z can be obtained from the standard normal table using the desired level of significance and the tail of test.  

The critical value tells the number of standards deviations away is the result from the mean.

Probability of type I error denoted by  a

Probability of type II error denoted by b.

Type I error is to falsely infer the existence of something that is not there, while a type II error is to falsely infer the absence of something that is.

From the given information,

a=0.01, b=0.05, o1 = o2 =0.05, Dο = 0, D' = 0.04

The sample size is calculated as,

n = [tex] ( o^2 + o2^2) * (Za +Zb)^2 /( (D' - D0)^2)  [/tex]

= [tex] (0.05^2 +0.05^2)*(2.33 + 1.645)^2 / (0.04 - 0)^2 [/tex]

= 49.38

which is 50 [Rounded to next integer]

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