Question 74645
<pre><font size = 4><b>
 Lengths of pregnancies of humans are normally distributed 
with a mean of 268 days and a standard deviation of 16 days. 
Use the Empirical Rule to determine the percentage of women 
whose pregnancies are between 252 and 284 days.


Calculate the following 6 values:

1. Calculate the mean minus 3 times the standard deviation, call it
   <font face = "symbol">m</font>-3<font face = "symbol">s</font>.

   268-3(16) = 220

2. Calculate the mean minus 2 times the standard deviation, call it
   <font face = "symbol">m</font>-2<font face = "symbol">s</font>.

   268-2(16) = 236

3. Calculate the mean minus the standard deviation, call it
   <font face = "symbol">m</font>-<font face = "symbol">s</font>.

   268-16 = 252

4. Calculate the mean plus the standard deviation, call it
   <font face = "symbol">m</font>+<font face = "symbol">s</font>.

   268+16 = 284

5. Calculate the mean plus 2 times the standard deviation, call it
   <font face = "symbol">m</font>+2<font face = "symbol">s</font>.

   268+2(16) = 300

6. Calculate the mean plus 3 times the standard deviation, call it
   <font face = "symbol">m</font>+3<font face = "symbol">s</font>.  

   268+3(16) = 316

[Shortcut hint, calculate the lowest one, 220 and then keep adding the
standard deviation 16 over and over)

Then line them up smallest to largest inserting the mean,
 <font face = "symbol">m</font>, itself in the middle.

<font face = "symbol">m-3s, m-2s, m-s, m, m+s, m+2s, m+3s</font>

220, 236, 252, 268, 284, 300, 316

Empirical Rule:

  0.15% of the data is below <font face = "symbol">m-3s</font>, i.e., below 220. 
  2.35% of the data is between <font face = "symbol">m-3s</font> and <font face = "symbol">m-2s</font>, ie., between 220 and 236.
 13.5 % of the data is between <font face = "symbol">m-2s</font> and <font face = "symbol">m-s</font>, i.e., between 236 and 252.
 34   % of the data is between <font face = "symbol">m-s</font> and <font face = "symbol">ms</font>, ie., between 252 and 268.
 34   % of the data is between <font face = "symbol">m</font> and <font face = "symbol">m+s</font>, ie., between 268 and 284.
 13.5 % of the data is between <font face = "symbol">m+s</font> and <font face = "symbol">m+2s</font>, i.e., between 284 and 300.
  2.35% of the data is between <font face = "symbol">m+2s</font> and <font face = "symbol">m+3s</font>, ie., between 300 and 316.
  0.15% of the data is above <font face = "symbol">m+3s</font>, i.e., above 316.
------
100.0%

to determine the percentage of women whose pregnancies 
are between 252 and 284 days, break it down into these 
groups:

34% are between 252 and 268, and 13.5% are between 268 
and 284, so that's a total of 34% + 13.5% or 47.5%.

That's the answer.

------------------------------------------

A shorter version of the empirical rule equivalent to 
the above is

For data sets having a normal distribution, the 
following properties apply:
About 68% of all values fall within 1 standard deviation 
of the mean.
About 95% of all values fall within 2 standard deviations
of the mean
About 99.7% of all values fall within 3 standard deviations
of the mean.

It's shorter to learn than the first, but harder to
calculate with.

Edwin</pre>