Question

When Samuel commutes to work, the amount of time it takes him to arrive is normally distributed with a mean of 32 minutes and a standard deviation of 5 minutes. Using the empirical rule, what percentage of his commutes will be between 17 and 47 minutes?

Answer 1

`z=(x-\mu)/(\sigma)`

And we can find the nnumber of deviations from the mean for each limit given:

`z_1 = (17-32)/(5) = -3`

`z_2= (47-32)/(5) = 3`

So we are 3 deviation from the mean and using the empirical rule we know that within 3 deviations from the mean we have 99.7% of the values

Explanation:

Let X the random variable that represent the amount of time it takes him to arrive, and for this case we know the distribution for X is given by:

`X \sim N(32,5)`

Where
`\mu=32` and
`\sigma=5`

We want to find this probability:

`P(17<X<41)`

And we can use the z score formula given by:

`z=(x-\mu)/(\sigma)`

And we can find the nnumber of deviations from the mean for each limit given:

`z_1 = (17-32)/(5) = -3`

`z_2= (47-32)/(5) = 3`

So we are 3 deviation from the mean and using the empirical rule we know that within 3 deviations from the mean we have 99.7% of the values