Question

A company tested 735 of the lightbulbs they produced and found them to have a mean life of 1,200 hours and a standard deviation of 50 hours. How many of these lightbulbs had a life between 1,170 hours and 1,230 hours?

Answer 1

We assume that the distribution of the values of life of the lightbulbs can be well approximated by a normal distribution with the same parameters: mean 1,200 hours and standard deviation 50 hours.

Then the value of life is equal to 1200 + 50X, where X is a standard normally distributed random variable. The probability that the value of life is between 1,170 and 1,230 hours is equal to the probability that the value of X is between -0.6 and 0.6:

$1170 \leq 1200 + 50X \leq 1230$

$(1170-1200)/50 \leq X \leq (1230-1200)/50$

$-0.6\leq X \leq 0.6$

This probability may be found as the integral

$\frac{1}{\sqrt{2\pi}}\int\limits_{-0.6}^{0.6}e^{-x^2/2}dx = \Phi(0.6) -\Phi(-0.6) =0.72575 - 0.27425 = 0.4515$

where $\Phi(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{x}e^{-t^2/2}dt$ - the cumulative distribution function of the standard normal distribution, its values we may get from the tables.

The estimated number of lightbulbs had a life between 1,170 hours and 1,230 hours is equal to $735 \cdot 0.4515 = 331.85$

Answer. The estimated number of lightbulbs had a life between 1,170 hours and 1,230 hours is equal to 331.85.