Answer:
2.28% of the batteries will fail within the first 65 months of use
Step-by-step explanation:
We have a bell shaped battery life distribution. Let X be the random variable that represents a battery life in months. If we suppose that we can model the battery lifes with the normal distribution with [tex]\mu = 75[/tex] months and [tex]\sigma = 5[/tex] months, we have the normal probability density function
[tex]f(x) = \frac{1}{\sqrt{2\pi(5)^{2}}}\exp[-\frac{(x-75)^{2}}{2(5)^{2}}][/tex],
we are seeking [tex]P(X \leq 65)[/tex].
[tex]P(X \leq 65) = \int\limits_{-\infty}^{65} f(x) dx[/tex] = 0.0228. So
2.28% of the batteries will fail within the first 65 months of use. We can use a table from a book or a programming language to compute the probability, here we use the instruction pnorm(65, mean = 75, sd = 5) and the R statistical programming language.
