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%? explain.
c. would you recommend further testing to determine whether the new parts have a lower failure rate than 4%? explain
a. the probability was about 40%
(round to the nearest percent as needed)
b. select the correct choice below, and fill in the answer box to complete your choice
(type an integer or a decimal)
○ a. the new component has a lower sample failure rate, % than the original, so it is reasonable to make that conclusion because the probability in part a is so low
○ b. the new component has a higher sample failure rate, %, than the original, so it is not a reasonable conclusion.
○ c. while the new component has a lower sample failure rate, %, than the original, it is not reasonable to make that conclusion based on the result from part a

Explanation:

Step1: Calculate sample failure rate

The sample failure rate is calculated as $\frac{\text{Number of defective items}}{\text{Total number of items tested}}$. Here, number of defective items is $1$ and total number of items tested is $50$. So the sample failure rate is $\frac{1}{50}=0.02 = 2\%$.

Step2: Analyze option A

Option A: The new component has a lower sample failure rate. The sample failure rate is $2\%$ (since $\frac{1}{50}=0.02$ or $2\%$). But the probability in part a (about $40\%$) is not “so low” (a probability of $40\%$ is a relatively common event).

Step3: Analyze option B

Option B: The new component has a higher sample failure rate. But $2\%<4\%$, so this option is incorrect.

Step4: Analyze option C

Option C: The new component has a lower sample failure rate ($2\%$). Since the probability in part a (about $40\%$) means that getting at most 1 defective out of 50 is a not - extremely - rare event (if the failure rate was still $4\%$). So we cannot be confident that the failure rate has actually decreased just based on this one test.

Answer:

C. While the new component has a lower sample failure rate, $2\%$, than the original, it is not reasonable to make that conclusion based on the result from part a.