Question

A normally-distributed data set has a mean of 30 and a standard deviation of 5.

What is the area under the standard normal curve for data greater than 25?

Answer 1

Let \(X\) denote the random variable for points in the data set. Then\[\mathbb P(X>25)=\mathbb P\left(\frac{X-30}{5}>\frac{25-30}{5}\right)=P(Z>-1)\]where \(Z\) follows the standard normal distribution.

The area you want to find corresponds to the one-directional probability that any given data point lies at least \(1\) standard deviation away from the mean. If you know the empirical rule, you know that approximately \(68\%\) of the distribution lies within \(1\) standard deviation from the mean, which means the remaining \(32\%\) lie outside of \(1\) standard deviation from the mean. Since the distribution is symmetric, half of this proportion lies to one side of the distribution at least \(1\sigma\) away from the mean, or about \(16\%\).