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Chapter 6

The Normal Distribution

David Chow

Oct 2014

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Learning Objectives

In this chapter, you will learn:

To compute probabilities from the normal

distribution

To determine whether a set of data is

approximately normal

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Importance of a Normal Dist

Many continuous variables seem to be normally

distributed

Many discrete variables can be approximated by a

normal distribution

Eg: The binomial distribution is symmetric when n is

large (more precisely, when n 5 and n(1-) 5)

By the central limit theorem, sampling distributions

are approximately normal (to be discussed in ch7)

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Properties

Bell-shaped & symmetric

By symmetry, = median = mode

Location is characterized by ,

Spread is characterized by .

The variable X has infinite range

I.e., - < X < +

In symbols, X ~ N (, 2).

Remark: f(X) =

f(X)

X

can take

any values.

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Probability and Area

The bell-shaped curve is called a density

function.

Probability of X is found by the

corresponding area under the density curve.

Hence, total area under the curve = 1

Such area (probability) can be found from

statistical tables in any statistics textbook.

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Probability and Area

Unlike discrete probability distributions, the probability of a particular value from a continuous distribution is zero.

Eg: P (download time = 4s) = 0

Reason 1: Probability is the area under the density curve.

Reason 2: A continuous variable has infinitely many possible values.

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Probability and Area

If X is continuous, a probability is meaningful if it

corresponds to a range (or an interval) of X.

Eg: P ( download time < 4.0s)

Eg: P (a X b), X = height

Whether it is

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Shape

By varying and , we obtain different normal distributions.

Eg: Waiting time (X) at two university train stations:

1. At Shatin Univ, = 6 min, = 1.8 min,

2. At Pokfulam Univ, = 5 min, = 1.5 min.

Identify (1) & (2) in the graph. What do they have in common?

= 1.35 min

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The Standardized Normal

Any normal distribution (with any and ) can be transformed into the standardized normal distribution (Z).

Transformation formula:

The random variable Z is also normally distributed, with = 0 and = 1

I.e., Z ~ N (0, 1).

Z = no. of std away from mean

XZ

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Eg: Waiting Time

2.050

100200

XZ

X = waiting time for customers at a bank

X is normally distributed with a mean of 100s and

standard deviation of 50s

X ~ N ( = 100, 2 = 502)

1. Find the Z-value for X = 200s

2. What is X if Z = -1.5?

Z = +2.0 means X = 200s is two std ____ mean

Z = -1.5 means X = ____

ANSWER

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Eg: Height https://www.youtube.com/watch?v=4R8xm19DmPM

Fig1: Suppose the height

of adult US females is

normally distributed

Mean = 162.2cm

Standard deviation = 6.8cm

Fig2: What is the

probability a randomly

selected female is

taller than 170.5cm?

Fig3: X to Z

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Example

Z

100

2.0 0

200 X ( = 100, = 50)

( = 0, = 1)

The transformation does not change the

shape, only the ______ has changed

The same distribution can be expressed

in original units (X), or

in standardized units (Z)

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Finding Normal Probability

Standardized normal distribution

Row: value of Z to the 1st decimal point

Column: value of Z to the 2nd decimal point

Cumulative standardized normal distribution

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Finding Normal Probability

The Cumulative Z-Table (attached) gives the probability of ____

To find P (a < X < b) where X ~ N (, 2),

Translate X-values to Z-values,

Check the required probability from the table

A visual check is often useful

X

Area Z

Summarizing this

chapter with a chart

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Eg1: P (Z < 2)

The value within the table gives the probability from Z = up to the desired Z value.

.9772

P (Z < 2.00) = .9772

Remember the empirical rule?

2.0

.

.

.

Z 0.00 0.01 0.02

0.0

0.1

Z 0 2.00

0.9772

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Eg2: Standardized Normal Distribution

a. Find the standard deviation of the normally distributed variable x.

b. What are the required probabilities?

ANSWER

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Eg3: Verify the Empirical Rule

range 6

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Eg4: Downloading

Time

X = the time it takes (in s) to download an image file, X ~ N ( = 8.0, 2 = 5.02).

Find P(X < 8.6)

X

8.6 8.0

= 8

= 5

Z 0.12 0

= 0

= 1

= 0.5478

= P(Z < 0.12)

P(X < 8.6)

Z .00 .01 .02

0.0 .5000 .5040 .5080

0.1 .5398 .5438 .5478

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

Next, find P(8.0 < X < 8.6)

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Eg4: Downloading Time (Find X Given the Probability)

Find X such that 20% of download

times are less than X.

First, use the table to find the Z-value of the

given probability of 0.20. Z = ____

Second, convert the Z-value to X units using

the transformation formula.

So 20% of the download times are ____. X ? 8.0

.2000

Z ? 0

Z . .03 .04 .05

-0.9 . .1762 .1736 .1711

-0.8 . .2033 .2005 .1977

-0.7 . .2327 .2296 .2266 80.3

0.5)84.0(0.8

ZX

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Assessing Normality

There are different ways to assess normality. For example,

1. Graphically, construct a histogram or a box-and-whisker plot.

2. Check the descriptive measures:

Do the mean, median and mode have similar values?

Is the range approximately 6?

3. Use the empirical rule:

About 67% of the observations lie within .

About 95% of the observations lie within 2.

4. A more precise alternative is to construct the normal probability plot

(not included)

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Review Questions: T or F

1. In a standard normal distribution, the probability that Z is greater than 0.5 is 0.5

2. In a standard normal distribution, the probability that Z is greater than 1.96 is 2

3. For a continuous random variable x, the probability density function f(x) represents the probability at a given value of x

4. Larger values of the standard deviation result in a normal curve that is shifted to the left

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Review Question

Source: Educator.com

https://www.youtube.com/watch?v=bYnIIZbeFes

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Appendix

Cumulative-Z

& Excel Commands

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Excel Command

FIND AREA

Find cumulative probability (i.e., area from the left) given X- or Z- values

=NORMDIST(X, , , true) returns the cumulative probability of a normal distribution. Eg: =NORMDIST(5, 4, 1, true)

gives P(X < 5, given = 4, = 1), I.e., 0.8413.

=NORMSDIST(Z) gives the

cumulative probability of a standardized normal. Eg: =NORMSDIST(0.12) gives

P(Z < 0.12).

FIND X-VALUES

Find X- or Z- values given the cumulative probability (i.e., area from the left)

=NORMSINV(cumulative probability) Eg: =NORMSINV(0.5)

=NORMINV(cumulative

probability, , )