Assume that a normal distribution of data has a mean of 14 and a standard deviation of 2. Use the 68−95−99.7 rule to find the percentage of values that lie below 20.

We are given a distribution with the mean `mu=14` and standard deviation `sigma=2.` We are told that the distribution is normal and follows the normal rule of 68–95–99.7. We are asked to find the percentage of values that lie below 20.

(See attachment)

In a normal distribution, approximately 68% of...

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We are given a distribution with the mean `mu=14` and standard deviation `sigma=2.` We are told that the distribution is normal and follows the normal rule of 68–95–99.7. We are asked to find the percentage of values that lie below 20.

(See attachment)

In a normal distribution, approximately 68% of the data values lie within one standard deviation of the mean; that is, no more than one standard deviation above or below the mean. Approximately 95% lie within two standard deviations of the mean and about 99.7% lie within three standard deviations of the mean.

We can draw out a normal curve and determine the approximate percentage of data values that lie in each section. If 68% lie within one standard deviation of the mean, then by symmetry 34% lie within one standard deviation below the mean and 34% lie within one standard deviation above the mean.

In the given situation, about 34% of data values lie between 12 and 14, and a further 34% lie between 14 and 16.

So one way to do this is to realize that 50% lie below the mean, so 50% of the data values are less than 14. (The mean, median, and standard deviation are the same in normal distributions.) Another 34% lie within one standard deviation above the mean (34% are between 14 and 16.) 13.5% are between 16 and 18. Another 2.35% are between 18 and 20.

So a total of 99.85% lie below 20.

We could also see that .0015 lie above 20 so 1-.0015=.9985 or 99.85% lie below.

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