Because of variability in the manufacturing process, the actual yielding point o

Question

Because of variability in the manufacturing process, the actual yielding point of a sample of mild steel subjected to increasing stress will usually differ from the theoretical yielding point. Let p denote the true proportion of samples that yield before their theoretical yielding point. If on the basis of a sample it can be concluded that more than 20% of all specimens yield before the theoretical point, the production process will have to be modified.

(a) If 12 of 44 specimens yield before the theoretical point, what is the P-value when the appropriate test is used? (Round your answer to four decimal places.)
P-value =

(b) If the true percentage of "early yields" is actually 50% (so that the theoretical point is the median of the yield distribution) and a level 0.01 test is used, what is the probability that the company concludes a modification of the process is necessary? (Round your answer to four decimal places.)

Explanation / Answer

Below are the null and alternate hypothesis
H0: p <= 0.2
H1: p > 0.2

pcap = 12/44 = 0.2727
value to test statistics
z = (0.2727 - 0.2)/sqrt(0.2*0.8/44) = 1.2056

p-value = 0.114

(B)
if pcap = 0.5

value to test statistics
z = (0.5 - 0.2)/sqrt(0.2*0.8/44) = 4.97

p-value = 0
As p-value is less than 0.01, we reject null hypothesis and there is need to modify the process.

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