Question

exercises

1. a travel agency did a survey and found that the average local family spends $1,900 on a summer vacation. the distribution is normally distributed with standard deviation $390.

a. what percent of the families took vacations that cost under $1,500? round to the nearest percent.
b. what percent of the families took vacations that cost over $2,800? round to the nearest percent.
c. find the amount a family would have spent to be the 60th percentile. round to the nearest dollar.

1. a distribution is normal with mean 60 and standard deviation 8. find the area of each of the following shaded regions to four decimal places.

a.
b.
c.
d.

Explanation:

Step1: Calculate z - score for part a

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 1900$, $\sigma=390$ and $x = 1500$.
$z=\frac{1500 - 1900}{390}=\frac{- 400}{390}\approx - 1.03$
Using the standard normal table, the area to the left of $z=-1.03$ is approximately $0.1515$, or $15\%$ when rounded to the nearest percent.

Step2: Calculate z - score for part b

For $x = 2800$, $\mu = 1900$ and $\sigma=390$.
$z=\frac{2800 - 1900}{390}=\frac{900}{390}\approx2.31$
The area to the right of $z = 2.31$ is $1 - 0.9896=0.0104$, or $1\%$ when rounded to the nearest percent.

Step3: Find z - score for 60th percentile for part c

The z - score corresponding to the 60th percentile from the standard - normal table is approximately $z = 0.25$.
Using the formula $x=\mu+z\sigma$, with $\mu = 1900$, $\sigma = 390$ and $z = 0.25$.
$x=1900+0.25\times390=1900 + 97.5\approx1998$

Step4: Calculate areas for problem 2

For part a:
$z=\frac{52 - 60}{8}=\frac{-8}{8}=-1$
The area to the left of $z=-1$ is $0.1587$.
For part b:
$z=\frac{64 - 60}{8}=\frac{4}{8}=0.5$
The area to the right of $z = 0.5$ is $1 - 0.6915=0.3085$.
For part c:
$z_1=\frac{48 - 60}{8}=\frac{-12}{8}=-1.5$
The area to the right of $z=-1.5$ is $1 - 0.0668 = 0.9332$.
For part d:
$z_1=\frac{48 - 60}{8}=-1.5$ and $z_2=\frac{80 - 60}{8}=\frac{20}{8}=2.5$
The area between $z=-1.5$ and $z = 2.5$ is $0.9938-0.0668 = 0.9270$.

Answer:

1.
a. $15\%$
b. $1\%$
c. $\$1998$
2.
a. $0.1587$
b. $0.3085$
c. $0.9332$
d. $0.9270$