you are a consultant to a company that manufactures components for cell phones. one of the components the…

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.\na. before the design improvements, what was the probability that, among 50 of the items, at most one of the items was defective?\nb. is it reasonable to conclude that the new components have a lower failure rate than 4%? explain.\nc. would you recommend further testing to determine whether the new parts have a lower failure rate than 4%? explain.

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.\na. before the design improvements, what was the probability that, among 50 of the items, at most one of the items was defective?\nb. is it reasonable to conclude that the new components have a lower failure rate than 4%? explain.\nc. would you recommend further testing to determine whether the new parts have a lower failure rate than 4%? explain.

Answer

Answer:

a. 0.6767 b. It is reasonable to suspect a lower failure - rate, but more evidence is needed. c. Yes, further testing is recommended.

Explanation:

Step1: Identify the distribution

The number of defective items follows a binomial distribution (X\sim B(n,p)), where (n = 50) and (p=0.04).

Step2: Calculate (P(X = 0))

The binomial probability formula is (P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}), where (C(n,k)=\frac{n!}{k!(n - k)!}). For (k = 0), (C(50,0)=\frac{50!}{0!(50 - 0)!}=1), (p = 0.04), (1-p=0.96), so (P(X = 0)=1\times0.04^{0}\times0.96^{50}\approx0.1326).

Step3: Calculate (P(X = 1))

(C(50,1)=\frac{50!}{1!(50 - 1)!}=50), (P(X = 1)=50\times0.04^{1}\times0.96^{49}\approx0.2741).

Step4: Calculate (P(X\leq1))

(P(X\leq1)=P(X = 0)+P(X = 1)\approx0.1326 + 0.2741=0.6767).

Step5: Analyze part b

In a sample of (n = 50), only 1 defective item was found. While this is less than what we would expect on average ((np=50\times0.04 = 2)), the sample size is relatively small. So, it is reasonable to suspect a lower failure - rate, but more evidence is needed.

Step6: Analyze part c

Since the sample size of 50 is relatively small and the result is just a single - point observation, further testing is recommended to get a more accurate estimate of the failure rate of the new components.