Question

you are a consultant to a company that manufactures components for cell phones. one of the components the company manufactures has a 4% failure rate. design changes have improved the quality of the component. a test of 50 of the new components found that only one of the new components is defective.
a. before the design improvements, what was the probability that, among 50 of the items, at most one of the items was defective?
b. is it reasonable to conclude that the new components have a lower failure rate than 4%?
c. would you recommend further testing to determine whether the new parts have a lower failure rate than 4%? explain.

c. choose the correct answer below.

a. the probability calculated in part a indicates that the old design would have been very unlikely to perform as well as the new design just due to random chance, so it is almost certain the decreased failure rate is repeatable, and no more testing is needed before switching to the new design
b. the probability calculated in part a indicates that that a test of 50 random components from the old design could have had the same results just due to chance. therefore, more testing is needed before deciding if the change in the failure rate is repeatable
c. although the probability calculated in part a indicates that a test of 50 random components from the old design could have had the same results just due to chance, further testing will not clarify whether the change in the failure rate was random or is repeatable. therefore, no more testing is recommended
d. the probability calculated in part a indicates that the old design would have been very unlikely to perform as poorly as the new design just due to random chance, so it is almost certain the decreased failure rate is repeatable, and no more testing is needed before scrapping the new design

Explanation:

Step1: Analyze option A

The old design has a 4% failure rate. A test of 50 components with at most 1 defective is not extremely unlikely for the old design. So, it's not almost certain the decreased failure rate is repeatable.

Step2: Analyze option B

The binomial probability formula for \(n = 50\), \(p=0.04\) (old - design failure rate), \(P(X = k)=\binom{n}{k}p^{k}(1 - p)^{n - k}\). \(P(X\leq1)=P(X = 0)+P(X = 1)=\binom{50}{0}(0.04)^{0}(0.96)^{50}+\binom{50}{1}(0.04)^{1}(0.96)^{49}\)

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A probability of about \(0.4\) (40%) means that a test of 50 random components from the old design could have had the same results (at most 1 defective) just due to chance. So, more testing is needed.

Step3: Analyze option C

Further testing can help distinguish between random variation and a real - change in the failure rate. For example, if we test a larger sample size \(n\) (using hypothesis - testing procedures like a one - sample proportion test \(H_0:p = 0.04\) vs \(H_1:p<0.04\)), we can get more evidence.

Step4: Analyze option D

As shown in the calculation for option B, the probability that the old design could perform as well (in terms of at - most 1 defective in 50) is not extremely low. So, it's not almost certain the decreased failure rate is repeatable.

Answer:

B. The probability calculated in part a indicates that a test of 50 random components from the old design could have had the same results just due to chance. Therefore, more testing is needed before deciding if the change in the failure rate is repeatable.