Question

3.1. For a randomly selected battery of this make, what is the probability that it will last between 30 and 34 months? 

Answer 1

Assume that the mean life of a particular brand of car battery is normally distributed with a mean of 28 months and a standard variation of 4 months.

Let $X=$ the mean life of a particular brand of car battery : $X\sim N(\mu, \sigma^2).$

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

Given $\mu=28, \sigma=4$

$$P(30<X<34)=P(X<34)-P(X\leq30)$$

$$=P(Z<\dfrac{34-28}{4})-P(Z\leq\dfrac{30-28}{4})$$

$$=P(Z<1.5)-P(Z\leq0.5)$$

$$\approx0.9331928-0.6914625\approx0.2417$$

The probability that a randomly selected battery  will last between 30 and 34 months is 0.2417.