A manufacturer of camping gear wishes to buy rivets whose breaking strength must

Question

A manufacturer of camping gear wishes to buy rivets whose breaking strength must exceed 5000 psi. Otherwise, the rivet is considered defective. The breaking strength of rivets is normally distributed. Manufacturer Rosie Riveter (RR) produces rivets with a mean breaking strength of 7000 psi and a standard deviation of 550 psi. Manufacturer Rivet Runs Through It (RT) produces rivets with a mean breaking strength of 6500 psi and a standard deviation of 390 psi.

a. Which manufacturer produces the lowest fraction of defective rivets?

b. Referring to the rivet problem, assume that the camping gear maker buys 80% of its rivets from the manufacturer with the lower fraction of defective rivets, and 20% of its rivets from the other manufacturer, and mixes them together. What is the probability that a rivet in the combined batch will be defective?

c.Referring to the rivet problem, suppose the manufacturer with the higher fraction of defective rivets can improve the process to adjust the standard deviation, but not the mean breaking strength. What new standard deviation should this manufacturer achieve to have the same probability of defective rivets as the other manufacture?

Explanation / Answer

a)

FIRST MANUFACTURER:

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    5000      
u = mean =    7000      
          
s = standard deviation =    550      
          
Thus,          
          
z = (x - u) / s =    -3.636363636      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -3.636363636   ) =    0.000138257 [ANSWER]

SECOND MANUFACTURER:

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    5000      
u = mean =    6500      
          
s = standard deviation =    390      
          
Thus,          
          
z = (x - u) / s =    -3.846153846      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -3.846153846   ) =    5.99932*10^-5

Hence, RT produces the lower fraction of defective rivets. [ANSWER]

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b)

Let

D = defective
RR = Rosie Riveter
RT = Rivet Runs

Hence,

P(D) = P(RR) P(D|RR) + P(RT) P(D|RT)

= 0.20*5.99932*10^-5 + 0.80*0.000138257

= 0.000122604 [ANSWER]

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C)

Hence, RR must have a z score of -3.846153846 for 5000 psi.

Hence,

sigma = (x-u)/z = (5000-7000)/(-3.846153846) = 520 psi [ANSWER]

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