Question

1. The machineries company that supplies machinery to the ship John claims that their bridge equipment has a mean life of 54 months with a standard deviation of 10 months. Assume that the variable is normally distributed.
a. If 40 randomly selected bridge equipment will be test, what is the probability that a randomly selected equipment will last for less than 50 months?
b. If 40 randomly selected bridge equipment will be test, what is the probability that it will for more than 56 months?

Answer 1

Let $X=$ service time of equipment: $X\sim N(\mu, \sigma^2/n ).$

Then $Z=\dfrac{X-\mu}{\sigma/\sqrt{n}}\sim N(0, 1)$

a) Given $\mu=54\ months, \sigma=10 \ months, n=40$

$$P(X<50)=P(Z<\dfrac{50-54}{10/\sqrt{40}})\approx$$

$$\approx P(Z<-2.529822)\approx0.005706$$

b) Given $\mu=54\ months, \sigma=10 \ months, n=40$

$$P(X>56)=1-P(X\leq 56)$$

$$=1-P(Z\leq\dfrac{56-54}{10/\sqrt{40}})\approx$$

$$\approx P(Z<1.264911)\approx1-0.897048$$

$$\approx0.102952$$