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An experiment was performed to compare the fracture toughness of high-purity 18 Ni maraging steel with...

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An experiment was performed to compare the fracture toughness of high-purity 18 Ni maraging steel with...

An experiment was performed to compare the fracture toughness of high-purity 18 Ni maraging steel with commercial purity stee
An experiment was performed to compare the fracture toughness of high-purity 18 Ni maraging steel with commercial purity steel of the same type. For m = 34 specimens, the sample average toughness was x = 63.1 for the high-purity steel, whereas for n = 39 specimens of commercial steel y = 57.3. Because the high- purity steel is more expensive, its use for a certain application can be justified only if its fracture toughness exceeds that of commercial-purity steel by more than 5. Suppose that both toughness distributions are normal. (a) Assuming that 01 = 1.2 and 02 = 1.1, test the relevant hypotheses using a = 0.001 5:51 PM (b) Compute B for the test conducted in part (a) when u1 - 2 = 6. (Round your answer to four decimal places.) 5:51 PM
Math Statistics And Probability
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Answers

H_0: \mu_x-\mu_y=5

H_a: \mu_x-\mu_y>5

A B с D E F H J K 1 SD N Mean 63.1 х 1.2 34 2 a) 3 Y 57.3 1.1 39 4 5 2.9533 ((C2-C3)-5)/SQRT((D2^2/E2)+(D342/E3)) 6 0.001 7 .

b) This is a right tailed test.

We will fail to reject the null (commit a Type II error) if we get a Z statistic less than 3.0902.( using excel formula =-NORM.S.INV(0.001))

This 3.0902 Z-critical value corresponds to some X critical value, such that

\frac{X_d-5}{\sqrt{\frac{1.2^2}{34}+\frac{1.1^2}{39}}}=3.0902

X_d=5.837

So I will incorrectly fail to reject the null as long as a draw a sample mean that is lesser than 5.837. To complete the problem what I now need to do is compute the probability of drawing a sample mean lesser than 5.837 given \mu_d=6 . Thus, the probability of a Type II error is given by

\beta=P(Z<\frac{5.837-6}{\sqrt{\frac{1.2^2}{34}+\frac{1.1^2}{39}}})=P(Z<-0.6017)=0.2737 ( using excel formula=NORM.S.DIST(-0.6017,TRUE) )

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