Page 166 - Statistics for Dummies

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Part III: Distributions and the Central Limit Theorem

The probability that X is equal to any single value is 0 for any continuous
random variable (like the normal). That’s because continuous random vari-
ables consider probability as being area under the curve, and there’s no area
under a curve at one single point. This isn’t true of discrete random variables.
Suppose, for example, that you enter a fishing contest. The contest takes
place in a pond where the fish lengths have a normal distribution with mean
μ = 16 inches and standard deviation σ = 4 inches.
✓ Problem 1: What’s the chance of catching a small fish — say, less than
8 inches?
✓ Problem 2: Suppose a prize is offered for any fish over 24 inches. What’s
the chance of winning a prize?
✓ Problem 3: What’s the chance of catching a fish between 16 and
24 inches?
To solve these problems using the steps that I just listed, first draw a pic-
ture of the normal distribution at hand. Figure 9-3 shows a picture of X’s
distribution for fish lengths. You can see where the numbers of interest
(8, 16, and 24) fall.
Problem 3
X
μ = 16
σ = 4
Problem 1
Figure 9-3:
Problem 2
The distribu-
tion of fish
lengths in a
4 8 12 16 20 24 28
pond.
Fish length (inches)
Next, translate each problem into probability notation. Problem 1 is really
asking you to find p(X < 8). For Problem 2, you want p(X > 24). And Problem 3
is looking for p(16 < X < 24).
Step 3 says change the x-values to z-values using the z-formula:
For Problem 1 of the fish example, you have the following:

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