Using the Empirical Rule to derive proportions

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to use the Empirical Rule to derive proportions.  Here's our problem statement: The blood platelet counts of a group of women have a bell-shaped distribution with a mean of 246.2 and a standard deviation of 68.8 (all units are 1000 cells per microliter).  Using the Empirical Rule find each approximate percentage below.  Part A: What is the approximate percentage of women with platelet counts within one standard deviation of the mean, or between 177.4 and 315.0.  Part B: What is the approximate percentage of women with platelet counts between 108.6 and 383.8?

Part A

So for this first part of the question, we want to know the approximate percentage of the data set that is within one standard deviation of the mean.  To do that, we can just use the Empirical Rule.  Here I have a graphic imported into Excel to help us understand the Empirical Rule.  So within one standard deviation of the mean is 68% of the data.  The key to using the Empirical Rule is that you have a bell-shaped distribution, which we clearly do here in the problem statement — “bell-shaped distribution.”  So in a bell-shaped distribution, one standard deviation of the mean contains 68% of the data.  So I'm just going to put 68 here.  Fantastic!

Part B

Now, approximately what percentage of women in this group have platelet counts between 108.6 and 383.8?  Well, this would be really easy if they gave us the number of standard deviations as they did in the previous part of the problem.  But here they don't do that.  So what we're going to have to do is figure out how many standard deviations are between these two numbers that they give us here.

To do that, I'm going to go to my calculator, subtract them to find the difference, and then divide by the standard deviation, which was listed here in the problem — “standard deviation of 68.8.”  So I'm going to divide by 68.8.  That gives me 4.  So that tells me that each one of these values that's given here is 2 standard deviations from the mean, because if I go back to my Empirical Rule, notice within two standard deviations of the mean I've got one, two, three, four standard deviation units.  So we're looking at 95% of the data in between those two numbers.  So I'll put 95 here.  Nice work!

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