Question

A new car was put into production. It involved many assembly tasks. Each car was inspected at the end of the assembly line and the number of defects per unit was recorded. For the first 100 cars produced, there were 40 defective cars. Some of the cars had no defects, a few had one defect, and so on. The distribution of defects followed a Poisson distribution. Based on the first 100 cars produced, about how many out of every 1,000 cars assembled should have one or more defects?

a. About 660
b. About 165
c. About 630
d. About 330

Answer 1

Option D is correct.

Out of every 1,000 cars assembled, the number of cars that should have one or more defects is about 330.

Explanation:

We can obtain the limits of the confidence interval for this problem and the mean must always be in that interval (the definition of confidence interval!)

Limits of the interval = (mean ± margin of error)

Mean = (sample size) × (proportion) = np

n = 1000

p = (40/100) = 0.4

Mean = 0.4 × 1000 = 400

Margin of error = (critical value) × (standard deviation of the sample mean)

Using a confidence interval of 99.999% to fully cover all grounds, critical value = z = 4.42

Standard deviation of the sample mean = √[np(1-p)] = √(1000×0.4×0.6) = 15.50

Margin of error = 4.42 × 15.50 = 68.51

Limits of the interval = (mean ± margin of error)

Limits of the interval = (400 ± 68.51)

Upper limit of the interval = 400 + 68.51 = 468.51

lower limit of the interval = 400 - 68.51 = 331.49

The confidence interval then is (331.49, 468.51).

From the options provided, about 330 is the closest to this confidence interval obtained.

Hope this Helps!!!