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Math Tech IIII, Mar 27
Finding Normal Distribution Probabilities
Book Sections: 5.1 & 5.2
Essential Questions: How can I compute probabilities using the
standard normal distribution, any normal distribution, and what do
those probabilities mean?
Standards: DA-4.7, DA-4.10, S.ID.4
Standard Normal Probabilities
• The area under the standard normal curve to the left of a zscore gives the probability that z is less than that z-score.
Example: The area to the left of z = -0.99 is 0.1611.
So P(z < -0.99) = 0.1611.
• You can equate probability with any area under the curve
before, after, or between any z-values. Here’s how:
Interpreting Standard Normal Probabilities
• P(z < z1) = area left of z1 (lookup procedure a)
• P(z > z1) = 1 – area left of z1 (which is the area right of z1)
(lookup procedure b)
• P(z1 < z < z2) = area left of z2 – area left of z1 (the area
between z1 and z2) (lookup procedure c)
• These computations are the ones from the last lesson, now
called ‘probability’ as a new interpretation.
Normal Distributions
Find the z for a given x
Any normal distribution with μ and σ
You can use the standard normal model to compute any
probability. The way in is through z-score and table.
Any Normal Distribution Probabilities
• There are an infinite number of normal distributions, but
only one Standard Normal Distribution.
• The standard normal distribution is a normal distribution
with a mean, of 0 and a standard deviation, of 1.
 Any distribution can be transformed here by the Z-Score.
z
(x  )

How It Works
The objective is to compute the probability of some
value with respect to the mean, μ and the standard
deviation, σ. You can normalize the computation
and apply the correct model to compute the desired
probability.
How do you normalize? Use the z-score formula.
The Step-by-Step Way
1. Read the problem, determine what is being asked.
Write an inequality.
2. Determine μ and σ, and any x-values in the problem.
3. Compute z score(s) needed.
4. Set up model based on step 1 for ‘This’ probability.
5. Enter table with z-score(s), extract probability value(s).
6. Compute probabilities based on model from step 4.
Example 1
The weights of adult male beagles are normally distributed with a
mean of 25 pounds and a standard deviation of 3 pounds. A beagle
is randomly selected. Compute each probability:
a) That the beagle’s weight is less than 23 pounds.
b) That the beagle’s weight is between 24 and 26 pounds.
c) That the beagle’s weight is more than 30 pounds.
Example 1a
The weights of adult male beagles are normally distributed with a
mean of 25 pounds and a standard deviation of 3 pounds. A beagle
is randomly selected. Compute each probability:
a) That the beagle’s weight is less than 23 pounds.
μ = 25, σ = 3
Example 1b
The weights of adult male beagles are normally distributed with a
mean of 25 pounds and a standard deviation of 3 pounds. A beagle
is randomly selected. Compute each probability:
b) That the beagle’s weight is between 24 and 26 pounds. μ = 25,
σ=3
Example 1c
The weights of adult male beagles are normally distributed with a
mean of 25 pounds and a standard deviation of 3 pounds. A beagle
is randomly selected. Compute each probability:
c) That the beagle’s weight is more than 30 pounds. μ = 25, σ = 3
Classwork: Handout CW 3/27, 1-10
Homework – None