by dfarris
Last Updated September 11, 2019 17:20 PM

Suppose the people living in a city have a mean score of $53$ and a standard deviation of $4$ on a measure of concern about the environment. Assume that these concern scores are normally distributed. Using the $50\%$-$34\%$-$14\%$ figures, what is the minimum score a person has to have to be in the top:

(a) $2\%$, (b) $16\%$, (c) $50\%$, (d) $84\%$, and (e) $98\%$?

$50$-$34$-$14$ is the empirical rule for calculating percentiles in a normal distribution. This means that $50\%$ of the data can be found on either side of the mean, $34\%$ of the data is found within one standard deviation on either side of the mean, and $14\%$ of the data can be found between one and two standard deviations on either side of the mean.

Algebraically, for a distribution $\mathcal{N}(\mu,\sigma^2)$:

- $50\%$ of the data can be found within the interval $(\mu-\sigma$, $\mu+\sigma)$;
- $34\%$ of the data can be found within the intervals $(\mu$, $\mu+\sigma)$ or $(\mu-\sigma$, $\mu)$; and
- $14\%$ of the data can be found within the intervals $(\mu+\sigma$, $\mu+2\sigma)$ or $(\mu-2\sigma$, $\mu-\sigma)$.

For your question, you have the distribution $\mathcal{N}(53,4^2)$. Use the above to figure out your answers, and share your solution.

Updated April 25, 2018 04:20 AM

Updated November 08, 2017 02:20 AM

Updated June 16, 2017 21:20 PM

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