Answer to Question #214982 in Statistics and Probability for Madimetja

Question

Most production processes generate defectives. However, the percentage of defectives will usually be kept as low as possible. To check the production process, random samples or products areusually drawn and tested.

Consider a certain mass-produced item. Suppose that the production process yields at most 2% defectives, at least when the process is what is termed ‘under control’. To check the quality of the production process, 400 of the items are randomly selected each hour. If at most 9 defectives are detected, then it is concluded that the process is, still under control and nothing is done. But if 10 or more defectives are found, then it is concluded that the production process is, in the terminology, ‘out of control’, and the process is stopped and investigated. Below, p denotes the proportion of defectives in the large hourly production and bp denotes the (random) sample proportion of the defectives in an hourly sample of size 400.

(a) Determine the probability distribution of p

Answer 1

The probability distribution of p under control:

$p= \frac{1}{400}, \frac{2}{400}, \frac{3}{400}, \frac{4}{400}, \frac{5}{400}, \frac{6}{400}, \frac{7}{400}, \frac{8}{400}, \frac{9}{400}$

Out of control: $p ≥\frac{10}{400}$