Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 7.8 years, and standard deviation of 0.7 years.

If you randomly purchase one item, what is the probability it will last longer than 6 years? (Give answer to 4 decimal places.)

Answer 1

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 7.8 years, and standard deviation of 0.7 years.

If one item is randomly purchased, to find the probability that it will last longer than 6 years.

Now, let X be the random variable denoting the lifespan of a randomly selected item.

Then, X follows normal distribution with mean 7.8 and standard deviation of 0.7.

So, we can say that Z=(X-7.8)/0.7 follows standard normal, ie. normal(0,1).

So, we have to find the probability that

Where, Z is the standard normal variate.

Where, phi is the distribution function of the standard normal variate.

So, the probability that the purchased item will last longer than 6 years is 0.9949.