Question

a set of average city temperatures in july are normally distributed with a mean of 23.5°c and a standard deviation of 2°c. the average temperature of rabat is 23°c. what proportion of average city temperatures are higher than that of rabat? you may round your answer to four decimal places.

Explanation:

Step1: Calculate the z - score

The formula for the z - score 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. Here, $\mu = 23.5$, $\sigma=2$, and $x = 23$. So, $z=\frac{23 - 23.5}{2}=\frac{- 0.5}{2}=-0.25$.

Step2: Find the proportion

We want to find $P(X>23)$, which is equivalent to $P(Z>- 0.25)$ in the standard normal distribution. Since the total area under the standard - normal curve is 1, and $P(Z > z)=1 - P(Z\leq z)$. Looking up the value of $P(Z\leq - 0.25)$ in the standard - normal table, we find that $P(Z\leq - 0.25)=0.4013$. Then $P(Z>-0.25)=1 - 0.4013 = 0.5987$.

Answer:

$0.5987$