Question

a set of average city temperatures in april are normally distributed with a mean of 19.7°c and a standard deviation of 2°c. the average temperature of cairo is 21.4°c. what proportion of average city temperatures are higher than that of cairo? 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, $\mu = 19.7$, $\sigma=2$, and $x = 21.4$. So, $z=\frac{21.4 - 19.7}{2}=\frac{1.7}{2}=0.85$.

Step2: Find the proportion of values higher than the z - score

We know that the total area under the normal - distribution curve is 1. We use the standard normal distribution table (or z - table) to find the cumulative distribution function $\Phi(z)$ which gives the proportion of values less than $z$. Looking up $z = 0.85$ in the z - table, we find that $\Phi(0.85)\approx0.8023$. The proportion of values higher than $z$ is $1-\Phi(z)$. So, the proportion of average city temperatures higher than that of Cairo is $1 - 0.8023=0.1977$.

Answer:

$0.1977$