Question

1. seattle, washington, averages $mu = 34$ inches of annual precipitation. assuming that the distribution of precipitation amounts is approximately normal with a standard deviation of $sigma = 6.5$ inches, determine whether each of the following represents a fairly typical year, an extremely wet year, or an extremely dry year.

a. annual precipitation of 41.8 inches
b. annual precipitation of 49.6 inches
c. annual precipitation of 28.0 inches

Explanation:

Step1: Calculate z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean, and $\sigma$ is the standard deviation. A z - score between - 2 and 2 is considered typical, $z> 2$ is extremely wet, and $z < - 2$ is extremely dry.

Step2: Calculate z - score for a

For $x = 41.8$, $\mu=34$, and $\sigma = 6.5$.
$z=\frac{41.8 - 34}{6.5}=\frac{7.8}{6.5}=1.2$
Since $-2<1.2<2$, it is a fairly typical year.

Step3: Calculate z - score for b

For $x = 49.6$, $\mu = 34$, and $\sigma=6.5$.
$z=\frac{49.6 - 34}{6.5}=\frac{15.6}{6.5}=2.4$
Since $z = 2.4>2$, it is an extremely wet year.

Step4: Calculate z - score for c

For $x = 28.0$, $\mu=34$, and $\sigma = 6.5$.
$z=\frac{28.0 - 34}{6.5}=\frac{-6}{6.5}\approx - 0.92$
Since $-2<-0.92<2$, it is a fairly typical year.

Answer:

a. Fairly typical year
b. Extremely wet year
c. Fairly typical year