Question

A quality control inspector is inspecting ball bearings made during a single day using a new process to determine the proportion of defective ball bearings. The inspector selects an SRS of 90 ball bearings from the 5000 ball bearings produced on that day. After close inspection of each ball bearing, 11 were found to be defective. a. Construct and interpret a 99% confidence interval for the true proportion of defective ball bearings among all ball bearings manufactured using the new process. (Use the 4-step process here: STATE, PLAN, DO, CONCLUDE). b. Using the old process, the proportion of defective ball bearings was 20%. The plant manager claims that the true proportion of defective ball bearings using the new process is different from 20%. Does your interval in part (a) suggest that this claim is plausible? Explain your reasoning с. A closer inspection of the ball bearings found 4 more defective ball bearings for a total of 15 defective ball bearings. Could this new total change your answer to part (b)? Support your answer with a calculation.

Answer 1

To construct a 99% confidence interval for the proportion of defective ball bearings made using the new process, we can use the formula for the confidence interval for a proportion.

Step-by-step explanation:

1. STATE: We want to construct a 99% confidence interval for the proportion of defective ball bearings made using the new process.
2. PLAN: Since we have a simple random sample (SRS) of 90 ball bearings from a larger population of 5000, we can use the formula for the confidence interval for a proportion.
3. DO: Using the formula, we calculate the confidence interval to be (0.023, 0.127).
4. CONCLUDE: We are 99% confident that the true proportion of defective ball bearings made using the new process is between 0.023 and 0.127.

b. The confidence interval in part (a) does not suggest that the claim by the plant manager is plausible. The confidence interval does not include the value of 20%, indicating that the true proportion of defective ball bearings using the new process is likely different from 20%.

c. The new total of 15 defective ball bearings would not change the answer to part (b). The confidence interval is based on the initial sample of 90 ball bearings, and adding more defective ball bearings does not change the proportion estimated by the original sample.