Stuck with Empirical Rule

Question

Hi,

I have spent alot of time trying to understand Empirical rule. All the examples show problems falling right into place.

Here is what I have

mean=69.5, standard deviation=2.4

___% between 64.7 and 74.3

if I do..69.5 + 2.4 =71.9

69.5 - 24= 67.1

69.5 + 2.4 + 2.4 =74.3

69.5 -2.4 -2.4 =2.15

is my answer 95% or something more precise?

what about % 69.5 and 74.3?

Answer 1

69.5 - 2 • 2.4 = 64.7, and 69.5 + 2 • 2.4 = 74.3. This means that the upper and lower boundaries are 2 standard deviations above and below the mean. If you are using the empirical rule, this is considered 95%. The actual percentage is more like 95.45%, but if you are told to use the empirical rule, then 95% is what you'll be using.

Now let's consider between 69.5 and 74.3. Since 69.5 is the mean, and the normal curve is symmetrical, and the upper limit is the same as in the previous problem, this will be half of the percentage/area from the previous problem. Half of 95% is 47.5%, and that is the percentage between 69.5 and 74.3.

More explanation of the empirical rule:

There is more than one way that the empirical rule can look, but all are equivalent. One way is to chop up the normal curve into 1-standard-deviation increments and designate the areas of each individual piece that's created. The piece that is between the mean and the first standard deviation is 34% (half of the 68% that are within 1 standard deviation of the mean), and between the first and second is 13.5% (34 + 13.5 = 47,5, which is half of the 95% that are within 2 standard deviations of the mean), and between the third and fourth is 2.35 (34 + 13.5 + 2.35 = 49.85, which is half of the 99.7% within 3 standard deviations of the mean). The left side of the normal curve is the mirror image of the right side, going down by standard deviation increments. I hope this helps.

Answer 2

The standard "empirical rule" looks like this:

In this case, the mean is 69.5 so that goes where the "0" is on the x-axis

1 standard deviation below that is 69.5 - 2.4 or 67.1 which goes where the "-1" is on the x-axis.

2 standard deviations below the mean is 69.5 - 2.4 - 2.4 or 64.7 which goes where the "-2" is on the x axis.

1 standard deviation above the mean is 69.5 + 2.4 or 71.9 which goes where the "+1" is on the x-axis.

2 standard deviations above the mean is 69.5 + 2.4 + 2.4 or 74.3 which goes where the "+2" is on the x axis.

So to find the probability from 2 standard deviations below the mean (64.7) to 2 standard deviation above the mean (74.3), we add 13.6 + 34.1 + 34.1 + 13.6 (from the graph) to get 95.4% or, rounded, 95%