A manufacturing company considers to use one of two types of materials based on

Question

A manufacturing company considers to use one of two types of materials based on the breaking strength. The sigma of the two materials are equal, and the value is 1.0 pounds per square inch (psi). Two random samples of different sizes are taken: n1= 10, and n2= 12. The sample means are calculated to be x-bar1= 162.5 and x-bar2= 155.0. The company will not use material type 1 unless its mean breaking strength exceeds that of material type 2 by at least 10 psi. open book, open notes, NO Internet (thus Nolaptops tablets oelphone. Colculators are allowed. The time for the test will be the entire session. Leave after returning the test Return the cover paper of the test with the question, having your name in the upperright corner. DO NOT WRITEON BACXOF PAGES You can work the problem starting on the cover page and you con continue on your own paper. All pages need to have your name in the upper comer, and MSE304 Testaz markedunder your name. the lower right corner pur the number of pages you turn in, what page out of what number of pages in all include in the total number of pages the cover page with the text of the problem, which will be marked 1 (page2 will be show all the steps, starting with theformula, then replacing by numerical values and intermediote computations you wishpontiol credit to be given to you. A manufacturing company considers to use one of two types of materials based on the breaking (psi). strength. The sigma of the two materials are equal, and of the value is 1.0 pounds per square inch to Two random samples of different sizes are taken: ni *10, and n2 12. The sample means are calculated ex-bari 2.5 and x-bar 155.0. The company will not use material type unless its mean breaking strength exceeds that of material type 2 by at least 10 psi a) Should the company use material type 1 based on the sample information (the decision will be made at a significance level of 0.05). (use Hypothesis Testing: Specify all steps, and the conclusion regarding Ho and a verbal statement). b) the P-value for part a). c) what is the 9 c on the difference in means, assuming that the difference in means is indeed 12psi? d) if alpha 0.05, what is the power of the test? e) The company wishes to find out if the sample sizes used (nn 10 and nz 12) are adequate to detect a difference of 12 psi.They will decide this based on calculating the needed sample size (considering equal sample sizes) using the data of the problem and the values calculated in the parts al to c). If this sample size is smaller than the sizes used then the sample sizes 10 and 12, are adequate. Calculate the needed sample size (n), and conclude if the initial sample sizes are adequate. (Make a brief verbal statement).

Explanation / Answer

Solution

Let X = Breaking strength of Type I material and Y = = Breaking strength of Type II material.

Let X ~ N(µ1, 12) and Y~ N(µ2, 22)

We are also given 1 = 2 = 2, say

Also given are: sample size for X,n1 = 10, for Y, n2 = 12,

sample means Xbar = 162.5 Ybar = 155

We must test if (µ1 =µ2) < 12 and if so company would go for Type 2 material.

H0: (µ1 - µ2) = 12   Vs   HA: (µ1 - µ2) < 12

Test Statistic: Z = [(Xbar - Ybar) - (µ1 - µ2)]/[{(1/n1) + (1/n2)}= (0.5)/{2.sqrt 0.1833)}

= 0.584

Distribution

Under H0, Z ~ N(0, 1).

p-value

Under H0, p-value = P(Z < 0.584) = 0.7204

Decision (Inference)

H0 is accepted at 5% level of significance since p-value of Zcal > 0.05

Conclusion

There is no evidence to suggest that mean breaking strength of Type 1 material exceeds that of

Type 2 material by less than 12 psi and hence the company should opt for Type 1 material.

Done

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