Question

1) The ages of trees in a forest are normally distributed with a mean of 25 years and a standard deviation of 5 years. Using the empirical rule, approximately what percent of the trees are between 20 and 30 years old?

2)Pizza delivery times at Pizza Time are normally distributed with a mean time of 27 minutes and a standard deviation of 3 minutes. Using the empirical rule, approximately what percent of pizzas are delivered between 24 and 30 minutes?

Answer 1

1) 68%

2) 68%

Explanation:

1) The ages of trees

We know the mean and the standard deviation.

The mean is:

`\mu=25`

The standard deviation is:

`\sigma=5`

The Z-score formula is:

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

For x=20 the Z-score is:

`Z_(20)=(20-25)/(5)=-1`

For x=30 the Z-score is:

`Z_(30)=(30-25)/(5)=1`

Then we look for the percentage of the data that is between
`-1 <Z <1` deviations from the mean.

According to the empirical rule 68% of the data is less than 1 standard deviations of the mean. This means that 68% of the trees are between 20 and 30 years old

2) Pizza delivery

First we calculate the Z-scores

We know the mean and the standard deviation.

The mean is:

`\mu=27`

The standard deviation is:

`\sigma=3`

The z-score formula is:

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

For x=24 the Z-score is:

`Z_(24)=(24-27)/(3)=-1`

For x=30 the Z-score is:

`Z_(30)=(30-27)/(3)=1`

Then we look for the percentage of the data that is between
`-1 <Z <1` deviations from the mean.

According to the empirical rule 68% of the data is less than 1 standard deviations of the mean. This means that 68% of pizzas are delivered between 24 and 30 minutes