The probability of a piston rod being of acceptable quality

The probability of a piston rod being of acceptable quality is p. From N such piston rods, the number of acceptable rods is X. The variable X has mean 258.99 and variance 7.77. Find the number of rods N and the probability (p) of a rod being acceptable.

Solution

prob for acceptable quality = p

Sample size = N

No of acceptable rods = x

Sample proportion = x/N

X has mean as 177.38 and variance 3.55

As being acceptable or not acceptable are two outcomes, and each piston is independent,
X is binomial

E(X) = np = 177.38

and var(x) = npq = 3.55

Hence q = 0.0200

p = 1-q = 0.9800

N = Mean/p = 181

No of rods = 181

and prob = 0.98