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

Answer:

1) 68%

2) 68%

Step-by-step 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:

$Z = \frac{x-\mu}{\sigma}$

For x=20 the Z-score is:

$Z_{20}=\frac{20-25}{5}=-1$

For x=30 the Z-score is:

$Z_{30}=\frac{30-25}{5}=1$

Then we look for the percentage of the data that is between $-1$ 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:

$Z = \frac{x-\mu}{\sigma}$

For x=24 the Z-score is:

$Z_{24}=\frac{24-27}{3}=-1$

For x=30 the Z-score is:

$Z_{30}=\frac{30-27}{3}=1$

Then we look for the percentage of the data that is between $-1$ 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