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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Macie Melton · Accepted Answer

We are given the information that this distribution is normal with a mean   μ of 180 and a standard deviation   σ of 20. This describes a distribution   N  (  180  ,      20    2    ).To answer the question, we will convert this problem into a standard normal distribution       N    s    (  0  ,      1    2    ) question by determining the z-scores for the interval 140-220. This can be done either by &quot;eyeballing it&quot; (since   σ is 20, and each endpoint of the interval is a multiple of   σ away from the mean   μ), or we can use the z-score formula:  z  =            x      −      μ        σ  Thus:      z          140        =            140      −      180        20    =  −      40    20    =  −  2        z          140        =            220      −      180        20    =      40    20    =  2  This tells us the interval we&#039;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 &quot;rule of thumb&quot; sometimes called the Empirical Rule which says the approximately 95% of the data in a normal distribution lies in the interval   [  −  2      σ    ,  2      σ    ], which is exactly what we&#039;re being asked. (The actual answer is more like 95.45%.)

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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