 # A set of data has a normal distribution with a mean of 180 and a standard deviation of 20. What percent of the data is in the interval 140 - 220? allucinemsj 2022-08-13 Answered
A set of data has a normal distribution with a mean of 180 and a standard deviation of 20. What percent of the data is in the interval 140 - 220?
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We are given the information that this distribution is normal with a mean $\mu$ of 180 and a standard deviation $\sigma$ of 20. This describes a distribution $N\left(180,{20}^{2}\right)$.
To answer the question, we will convert this problem into a standard normal distribution ${N}_{s}\left(0,{1}^{2}\right)$ question by determining the z-scores for the interval 140-220. This can be done either by "eyeballing it" (since $\sigma$ is 20, and each endpoint of the interval is a multiple of $\sigma$ away from the mean $\mu$), or we can use the z-score formula:
$z=\frac{x-\mu }{\sigma }$
Thus:
${z}_{140}=\frac{140-180}{20}=-\frac{40}{20}=-2\phantom{\rule{0ex}{0ex}}{z}_{140}=\frac{220-180}{20}=\frac{40}{20}=2\phantom{\rule{0ex}{0ex}}$
This tells us the interval we're being asked about is analogous to determining what percent of the standard normal distribution ${N}_{s}$ lies between z-scores of -2 and 2.
In statistics there is a handy "rule of thumb" sometimes called the Empirical Rule which says the approximately 95% of the data in a normal distribution lies in the interval $\left[-2\sigma ,2\sigma \right]$, which is exactly what we're being asked. (The actual answer is more like 95.45%.)