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ES

xa b

P a x b( )

Chapter 3 ~ Normal Probability Distributions

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ES Chapter Goals

• Learn about the normal, bell-shaped, or Gaussian distribution

• How probabilities are found

• How probabilities are represented

• How normal distributions are used in the real world

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ES Normal Probability Distributions

• The normal probability distribution is the most important distribution in all of statistics

• Many continuous random variables have normal or approximately normal distributions

• Need to learn how to describe a normal probability distribution

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ES Normal Probability Distribution

1. A continuous random variable

2. Description involves two functions:a. A function to determine the ordinates of the graph picturing the distributionb. A function to determine probabilities

4. The probability that x lies in some interval is the area under the curve

3. Normal probability distribution function:

This is the function for the normal (bell-shaped) curvef x e

x

( )( )

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2

2

1

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ES

2 3 2 3

The Normal Probability Distribution

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ES

• Illustration

P a x b f x dxa

b( ) ( )

a b x

Probabilities for a Normal Distribution

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ES Notes The definite integral is a calculus topic We will use a table to find probabilities for normal distributions

We will learn how to compute probabilities for one special normal distribution: the standard normal distribution

Transform all other normal probability questions to this special distribution

Recall the empirical rule: the percentages that lie within certain intervals about the mean come from the normal probability distribution

We need to refine the empirical rule to be able to find the percentage that lies between any two numbers

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ES Percentage, Proportion & Probability

• Basically the same concepts

• Percentage (30%) is usually used when talking about a proportion (3/10) of a population

• Probability is usually used when talking about the chance that the next individual item will possess a certain property

• Area is the graphic representation of all three when we draw a picture to illustrate the situation

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ES The Standard Normal Distribution

• There are infinitely many normal probability distributions

• They are all related to the standard normal distribution

• The standard normal distribution is thenormal distribution of the standard variable z(the z-score)

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ES Standard Normal DistributionProperties:• The total area under the normal curve is equal to 1• The distribution is mounded and symmetric; it extends indefinitely in both

directions, approaching but never touching the horizontal axis• The distribution has a mean of 0 and a standard deviation of 1• The mean divides the area in half, 0.50 on each side• Nearly all the area is between z = -3.00 and z = 3.00

Notes: Appendix, Table A lists the probabilities associated with the intervals from

the mean (0) to a specific value of z Probabilities of other intervals are found using the table entries, addition,

subtraction, and the properties above

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ES Which Table to Use?

An infinite number of normal distributions means an infinite number of tables to look

up!

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ES Solution: The Cumulative Standardized Normal Distribution

Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.5478.02

0.1 .5478

Cumulative Standardized Normal Distribution Table (Portion)

Probabilities

Shaded Area Exaggerated

Only One Table is Needed

0 1Z Z

Z = 0.120

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ES Standardizing Example

6.2 5 0.1210

XZ

Normal Distribution

Standardized Normal

Distribution

Shaded Area Exaggerated

10 1Z

5 6.2 X Z0Z

0.12

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ES Example:

Normal Distribution

Standardized Normal

Distribution

Shaded Area Exaggerated

10 1Z

7.1 X Z0Z 0.21

2.9 5 7.1 5.21 .2110 10

X XZ Z

2.9 0.21

.0832

2.9 7.1 .1664P X

.0832

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Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.5832.02

0.1 .5478

Cumulative Standardized Normal Distribution Table (Portion)

Shaded Area Exaggerated

0 1Z Z

Z = 0.21

Example: 2.9 7.1 .1664P X (continued

)

0

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Z .00 .01

-03 .3821 .3783 .3745

.4207 .4168

-0.1.4602 .4562 .4522

0.0 .5000 .4960 .4920

.4168.02

-02 .4129

Cumulative Standardized Normal Distribution Table (Portion)

Shaded Area Exaggerated

0 1Z Z

Z = -0.21

Example:

2.9 7.1 .1664P X (continued)

0

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ES Notes The symmetry of the normal distribution is a key factor in determining

probabilities associated with values below (to the left of) the mean. For example: the area between the mean and z = -1.37 is exactly the same as the area between the mean and z = +1.37.

When finding normal distribution probabilities, a sketch is always helpful. See course worksheet.

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ES Example:

8 .3821P X

Normal Distribution

Standardized Normal

Distribution

Shaded Area Exaggerated

10 1Z

5 8 X Z0Z

0.30

8 5 .3010

XZ

.3821

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ES Example:

8 .3821P X (continued)

Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.6179.02

0.1 .5478

Cumulative Standardized Normal Distribution Table (Portion)

Shaded Area Exaggerated

0 1Z Z

Z = 0.300

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ES Recovering X Values for Known Probabilities

5 .30 10 8X Z

Normal Distribution

Standardized Normal

Distribution10 1Z

5 ? X Z0Z 0.30

.3821.6179

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ES Assessing Normality• Not all continuous random variables are normally

distributed• It is important to evaluate how well the data set

seems to be adequately approximated by a normal distribution

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ES Assessing Normality

• Construct charts– For small- or moderate-sized data sets, do stem-and-

leaf display and box-and-whisker plot look symmetric?

– For large data sets, does the histogram or polygon appear bell-shaped?

• Compute descriptive summary measures– Do the mean, median and mode have similar

values?– Is the interquartile range approximately 1.33 ?– Is the range approximately 6 ?

(continued)

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ES Assessing Normality

• Observe the distribution of the data set– Do approximately 2/3 of the observations lie

between mean 1 standard deviation?– Do approximately 4/5 of the observations lie

between mean 1.28 standard deviations?– Do approximately 19/20 of the observations lie

between mean 2 standard deviations?

(continued)

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ES

• With Minitab

Assessing Normality

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ES Applications of Normal Distributions

• Apply the techniques learned for the z distribution to all normal distributions

• Start with a probability question in terms ofx-values

• Convert, or transform, the question into an equivalent probability statement involvingz-values

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ES

c x

0 c z

Standardization• Suppose x is a normal random variable with mean and

standard deviation

• The random variable has a standard normal distribution

z x

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ES Notes• The normal table may be used to answer many kinds of questions

involving a normal distribution

Example: The waiting time x at a certain bank is approximately normally distributed with a mean of 3.7 minutes and a standard deviation of 1.4 minutes. The bank would like

to claim that 95% of all customers are waited on by a teller within c minutes. Find the value of c that makes this statement true.

• Often we need to find a cutoff point: a value of x such that there is a certain probability in a specified interval defined by x

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ES

P x c

P x c

P z c

( ) 0..

..

.0.

..

0.

953 7

143 7

1495

3 714

95

c

cc

3714

1645

1645 14 3 7 6 0036

..

.

( . )( . ) . . minutes

3 7. c x0 1645. z

0 5000. 0 4500.

0 0500.

Solution

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ES Notation• If x is a normal random variable with mean and standard deviation , this is often

denoted:x ~ N(, )

Example: Suppose x is a normal random variable with = 35 and = 6. A convenient notation to identify this random variable is: x ~ N(35, 6).

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ES Notation

• z-score used throughout statistics in a variety of ways

• Need convenient notation to indicate the area under the standard normal distribution

• z() is the algebraic name, for the z-score (point on the z axis) such that there is of the area (probability) to the right or left of z()

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ES Notes• The values of z that will be used regularly come from

one of the following situations:

1. The z-score such that there is a specified area in one tail of the normal distribution

2. The z-scores that bound a specified middle proportion of the normal distribution

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ES Example Example: Find the numerical value of z(0.99):

• Because of the symmetrical nature of the normal distribution, z(0.99) = -z(0.01)

• Using Table A: z(0.01) = -2.33

0 z

0.01

z(0.01)

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ES Example Example: Find the z-scores that bound the middle 0.99 of the

normal distribution:

• Use Table A:z(0.005) = -2.575 and z(0.995) = 2.575

0

0 495.0 495.0 005.0 005.

z(0.005) z(0.995)

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ES Chapter Summary

• Discussed the normal distribution

• Described the standard normal distribution

• Evaluated the normality assumption