Section 1.3: The Normal Distribution - Purdue University

Section 1.3: The Normal Distribution

Learning goals for this chapter:

Know when and how to use the empirical (68-95-99.7% rule). Understand what the standard Normal distribution is and how it is related to other Normal distributions. Calculate both forwards and backwards Normal distribution problems. Draw a Normal distribution curve appropriate for a story.

Normal curves ? N( , ) ? Bell-shaped, unimodal, symmetric ? Mean, , always in the center of the curve (the peak) ? Standard deviation, , controls the spread of the graph (wide or narrow) ? Probabilities are just the area under the curve (integral) between the points of interest ? Total area under the Normal curve = 1 (or 100%) ? Curve stretches from - to + , but the area under the curve gets very small the farther you go from the mean.

In the last set of Chapter 1 notes, we discussed bell-shaped distributions, standardization, and the 68-95-99.7% rule.

Standard Normal Distribution

What if you need different probabilities for X N( , )? Do we have to use Calculus? No. We have a great shortcut--the Normal table, Table A, in the front cover of your book.

You must convert X N( , ) to Z N(0, 1), where Z has the standard Normal distribution. Convert using the formula:

Z x

Z-scores are what you need in order to use Table A in the front cover of your book.

Z-scores also let you compare 2 values from different Normal distributions to see their probabilities on the same scale.

P (Z < z-score) is what you will find on the Normal table. What if you want to know something else?

P(Z > z-score) = 1 ? P(Z z-score)

P(Z = z-score) = 0. Therefore

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P(Z z score) P(Z z score) and P(Z z score) P(Z z score) P(a < Z < b) = P(Z < b) ? P(Z < a). Z-scores tell you how far (measured in standard deviations) the original observations fall from the mean. To find a probability if you have X N( , ) and a sample score to work with: 1. Convert X to Z. Z x 2. Rearrange (if necessary) the inequality so that it uses < or . Remember that P(Z > z-score) = 1 ? P(Z z-score). 3. Look up the probability for your z-score on Table A. 4. If z-score is between 2 table values, either pick the closer one or average the two closest values. Example: Checking account balances are ~ N(1325, 25). Bill has a balance of $1270. a) What is the probability an account will have less money than Bill's?

b) What is the probability an account balance will be more than $1380?

c) What is the probability an account balance will be exactly $1380?

d) What is the probability an account will have less than $1325 (the mean)?

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e) What is the probability that an account will have between $1310 and $1390?

f) What is the probability an account will have less than $10?

"Backwards" Normal Problems If you are given the probability and know X N( , ), but you don't know the sample's score (backwards from the previous problems): 1. Treat it as P(Z < z-score) = the probability. Work backward from the probability

in Table A to a corresponding z-score. 2. Adjust to < if necessary by doing the 1 ? trick. 3. If you have a 2-sided probability, use P(-z0 < Z < z0) = 2 P(Z < z-score) ? 1. 4. Convert the z-score to x by converting with x = + z . Example: In the checking account example where the balances are ~ N (1325, 25),

a) What is the account balance, x0, such that the percentage of balances less than it is 23%?

b) What is the account balance, x0, such that the probability of a balance being more than it is 0.15?

c) Between what 2 central values do 40% of the balances fall?

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Mixed forwards and backwards problems: IQ Scores are Normally distributed with a mean of 100 and a standard deviation of 15.

1. What is the IQ range for the top 10% of people? 2. What percent of the population scores between 100 and 120? 3. Between what two scores does the central 20% of the population have? 4. If a person is selected at random, what is the chance he scores below an 85?

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