Solving Empirical Rule Questions

The Empirical Rule is an ESTIMATE, so you shouldn't use it unless a question specifically asks you to solve using the Empirical (or 68-95-99.7) Rule.

Steps to Solving Empirical Rule Questions
  1. Draw out a normal curve with a line down the middle and three to either side.
  2. Write the values from your normal distribution at the bottom. Start with the mean in the middle, then add standard deviations to get the values to the right and subtract standard deviations to get the values to the left.
  3. Write the percents for each section (you will need to memorize them!) 0.15, 2.35, 13.5 and 34.
  4. Determine the section of the curve the question is asking for and shade it in.
  5. Add up the percents in the sections that got shaded.
Note: If you get a value in (4) that is not one your wrote down in (2), then you either made a simple math error or you aren't supposed to use the Empirical Rule!

Example: The weights of adorable, fluffy kittens are normally distributed with a mean of 3.6 pounds and a standard deviation of 0.4 pounds. Answer the following questions, using the Empirical Rule. 

First, draw your Empirical curve with the 4 percentages! (Steps 1-3 are completed below.)



What percent of adorable, fluffy kittens weigh between 2.8 and 4.8 pounds? 

Step 4: We need to shade the region they are asking for.

Step 5: We need to add the percents in the shaded areas.

    13.5% + 34% + 34% + 13.5% + 2.35%  = 97.35%

What percent of adorable, fluffy kittens weigh less than 2.4 pounds? 

Step 4: We need to shade the region they are asking for.

Step 5: We need to add the percents in the shaded areas.

    0.15%

What value corresponds to a 97.5th percentile of kitten weights? 

This problem is a little different, they've given us the % and what us to find the weight that marks that cutoff. We need to determine first if the kitten weight is something above or below the mean. Since the 97.5 percentile means the kitten is fatter than 97.5% of kittens, we know it's ABOVE the mean. You have three values to check!

Step 4: Let's say you guess 4 pounds first, let's shade the area.

Step 5: You'd then just add up the percents and see if this was right.

0.15% + 2.35% + 13.5% + 34% + 34%   = 84%

Note: You didn't really need to add up the first four numbers - they are the first HALF of the curve, so you could have saved time doing the following: 
50% + 34% = 84%

Which is not what we're looking for. So we'd try 4.4 next:
Step 5: We don't have to start completely from stratch. It's what we had previously, with another 13.5% added.
84% + 13.5% = 97.5% 

So 4.4 pounds is the kitten weight that marks the 97.5th percentile.




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