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Warm-up March 18, 2013

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- Warm-up March 18, 2013

68% of the area under the bell curve is within ONE standard deviation of the mean.

68% Area

68% Area

95% of the area under the bell curve is within TWO standard deviations of the mean.

95% Area

95% Area

The bell curve is symmetric.

This means 50% of the area is to the right of the mean, 34% between m and m+s, and 47.5% between m and m-2s.

50% area

68/2 = 34% area

95/2 = 47.5% area

We could separate the bell curve into six “chunks”, with areas shown below.

Again, the area within ONE s.d. of the mean is

.34 + .34 = .68

Area within TWO s.d.’s of the mean is .135+.34+.34+.135

= .95

.34

.34

.025

.025

.135

.135

Now we can find any normal probability involving the mean plus or minus one or two standard deviations.

Example: Suppose that

X ~ N(6,9)

What is P(3<X<12)?

m = 6

s2 = 9 →s = 3

Here is the picture that should pop up in your head:

The probability of being between one s.d. below the mean and two s.d.’s above is:

.34 + .34 + .135

= .815

0

3

6

9

12