Learning Objectives

1. Find and interpret the area under a normal curve

2. Find the value of a normal random variable

Standardizing a Normal Random Variable

If a random variable X is normally distributed with a Mean μ and Standard Deviation σ, then the random variable is normally distributed with a Mean (μ) of 0 and a SD (σ) of 1. The random variable Z is said to have the standard normal distribution.

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Standard Normal Curve

Area under the standard normal curve are values to the left of a specified Z-score, z, as shown in the figure.

EXAMPLE IQ scores can be modeled by a normal distribution with μ = 100 and σ = 15.

How many standard deviations is an individual with an IQ score of 120 above the mean?

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  • An individual whose IQ score is 120 is 1.33 SDs above the mean.

How do you find area under standard normal curve to the left of a z = 1.33

• You use this chart:

• The area under the standard normal curve to the left of z = 1.33 is 0.9082.

  • the probability that a person has an IQ less than or equal to 120 is 90.82%.

    • You may alternatively say that a person with an IQ of 120 is at about the 91^st percentile.

Use the Complement Rule to find the area to the right of z = 1.33.

Areas Under the Standard Normal Curve'

EXAMPLE Finding the Area Under the Standard Normal Curve to the Left of Z

(a) Find the area under the standard normal curve to the left of z = −0.38.

(a) area to left of a z = -0.38 is 0.3520

EXAMPLE Finding the Area Under the Standard Normal Curve to the Right of Z

(a) Find the area under the standard normal curve to the right of z = 1.25.

area under the normal curve to the right of z is found by 1 − Area to the left of z_.

• 1 - area to left of 1.25

• 1 - 0.8944 = 0.1056

(a) Area to right of z = 1.25 is 0.1056