Question

A manufacturer of metal pistons claims that on average, about 10% of their pistons are defective (either oversize or undersize). This month, there are 100,000 pistons produced. The quality control staffs randomly selected 150 pistons to examine the size, and 20 pistons are found to defective? (No points will be given without proper steps) (1) (1pt) If we define variable x = number of defective pistons out of the 150 pistons selected, what distribution does x follow? (2) (1pt) Check on the conditions of normal approximation to binomial, is it okay to use normal distribution to approximate? (3) (4pts) From Normal approximation (don't forget about the continuity correction factor), what is the probability to observe 20 or more defective pistons out of the 150 randomly selected?

Answer 1

(a) x follows the binomial distribution.

(b) n = 150

p = 0.10

n*p = 150*0.10 = 15

n*(1 - p) = 150*(1 - 0.10) = 135

Since n*p > 5 and n*(1 - p) > 5, so we can use the normal approximation to the binomial distribution.

(c)   $\sigma$ = $\sqrt{}$ 150*0.10*(1 - 0.10) = 3.67

z = (19.5 - 15)/3.67 = 1.22

P (z $\geq$ 1.22) = 0.1103