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
m-3s.
268-3(16) = 220
2. Calculate the mean minus 2 times the standard deviation, call it
m-2s.
268-2(16) = 236
3. Calculate the mean minus the standard deviation, call it
m-s.
268-16 = 252
4. Calculate the mean plus the standard deviation, call it
m+s.
268+16 = 284
5. Calculate the mean plus 2 times the standard deviation, call it
m+2s.
268+2(16) = 300
6. Calculate the mean plus 3 times the standard deviation, call it
m+3s.
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,
m, itself in the middle.
m-3s, m-2s, m-s, m, m+s, m+2s, m+3s
220, 236, 252, 268, 284, 300, 316
Empirical Rule:
0.15% of the data is below m-3s, i.e., below 220.
2.35% of the data is between m-3s and m-2s, ie., between 220 and 236.
13.5 % of the data is between m-2s and m-s, i.e., between 236 and 252.
34 % of the data is between m-s and ms, ie., between 252 and 268.
34 % of the data is between m and m+s, ie., between 268 and 284.
13.5 % of the data is between m+s and m+2s, i.e., between 284 and 300.
2.35% of the data is between m+2s and m+3s, ie., between 300 and 316.
0.15% of the data is above m+3s, 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
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