Question

A set of city temperatures in April are normally distributed with a mean of 19.7 degrees Celsius and a standard deviation of 2 degrees Celsius. The average temperature in Cairo is 21.4 degrees Celsius.

What percentage of average city temperatures are higher than that of Cairo?

Answer 1

19.77% of average city temperatures are higher than that of Cairo

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

`\mu = 19.7, \sigma = 2`

What percentage of average city temperatures are higher than that of Cairo?

This is 1 subtracted by the pvalue of Z when X = 21.4.

`Z = (X - \mu)/(\sigma)`

`Z = (21.4 - 19.7)/(2)`

`Z = 0.85`

`Z = 0.85` has a pvalue of 0.8023

1 - 0.8023 = 0.1977

19.77% of average city temperatures are higher than that of Cairo