brightdim wrote:

A certain brand of automobile tire has a mean life span of 35,000 miles and has a standard deviation of 2250 miles. (Assume the life spans of the tires have a bell-shaped distribution).

The life spans of three randomly selected tires are 30,500 miles, 37,250 miles, and 35,000 miles. Using the Empirical Rule, find the percentile that corresponds to each life span.

I understand how to solve using the Empirical Rule, but I don't understand how to use that answer to figure out the percentile it goes with. I'm hoping someone can explain that to me.

You are to convert the percentages in the Empirical Rule to percentiles. In the following graph of a bell-shaped curve, the mean is

and the standard deviation is

What percentile is at

? This percentile is the percentage of the area under the curve to the left of

Using the Empircal Rule, 95% of the area under the curve is between the two lines at

and

so the area outside those two lines is

Half of this 5% is to the left of the line at

which means the percentile at

is

The percentile at

is the sum of the areas to the left the line at

which is