Answer to Question #150568 in Statistics and Probability for Jim

Question

A manufacturer of light bulbs finds that one light bulb model has a
mean life span of 1025 h with a standard deviation of 87 h. What percent of these
light bulbs will last?
a. at least 950 h?
b.
between 800 and 900 h?

Answer 1

Let $X$ be a life span of a bulb: $X\sim N(\mu, \sigma^2)$

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

Given $\mu=1025\ h,\sigma=87\ h$

a)

$$P(X\geq950)=1-P(X<950)$$

$$=1-P(Z<\dfrac{950-1025}{87})\approx1-P(Z<0.862069)$$

$$\approx1-0.194325=0.805675$$

b)

$$P(800<X<900)=P(X<900)-P(X\leq800)$$

$$=P(Z<\dfrac{900-1025}{87})-P(Z<\dfrac{800-1025}{87})$$

$$\approx P(Z<-1.436782)-P(Z<-2.586207)$$

$$\approx0.075390-0.004852=0.070538$$