Question

A battery lasts, average 4 years with a standard deviation of 1.4 years.the battery life is normally distributed

Answer 1

Let $X=$the lifetime of a battery: $X\sim N(\mu, \sigma^2)$

Given $\mu=4, \sigma=1.4$

a)

$$P(X<3)=P(Z<\dfrac{3-\mu}{\sigma})=P(Z<\dfrac{3-4}{1.4})$$

$$\approx P(Z<-0.7143)\approx0.2375$$

b)

$$P(1.8<X<5)=P(X<5)-P(X\leq1.8)$$

$$=P(Z<\dfrac{5-4}{1.4})-P(Z\leq\dfrac{1.8-4}{1.4})$$

$$\approx P(Z<0.7143)-P(Z\leq -1.5714)$$

$$\approx0.76248-0.05804\approx0.7044$$

c)

$$P(X<x)=P(Z<\dfrac{x-4}{1.4})=0.25$$

$$\dfrac{x-4}{1.4}\approx-0.6745$$

$$x\approx4-1.4(0.6745)$$

$$x\approx3.0557$$