Download Lecture 23 - The Normal Distribution

The Normal
Distribution
Lecture 23
Section 6.3.1
Tue, Feb 27, 2007
The “68-95-99.7 Rule”

For any normal distribution,
 Approximately
68% of the values lie within
one standard deviation of the mean.
 Approximately 95% of the values lie within
two standard deviations of the mean.
 Approximately 99.7% of the values lie within
three standard deviations of the mean.
The Empirical Rule


The well-known Empirical Rule is similar, but
more general.
If X has a “mound-shaped” distribution, then
 Approximately
68% lie within one standard deviation
of the mean.
 Approximately 95% lie within two standard deviations
of the mean.
 Nearly all lie within three standard deviations of the
mean.
Example
Use Excel to generate 1000 random
numbers with a normal distribution.
 Count the number of values that fall within
the three intervals.
 EmpiricalRule.xls.

The Standard Normal
Distribution
The standard normal distribution
 It is denoted by the letter Z.
 That is, Z is N(0, 1).

The Standard Normal
Distribution
N(0, 1)
z
-3
-2
-1
0
1
2
3
Areas Under the Standard
Normal Curve
What is the total area under the curve?
 What proportion of values of Z will fall
below 0?
 What proportion of values of Z will fall
above 0?

Areas Under the Standard
Normal Curve




What proportion of values will fall below +1?
What proportion of values will fall above +1?
What proportion of values will fall below –1?
What proportion of values will fall between –1
and +1?
Areas Under the Standard
Normal Curve

It turns out that the area to the left of +1 is
0.8413.
0.8413
z
-3
-2
-1
0
1
2
3
Areas Under the Standard Normal
Curve

So, what is the area to the right of +1?
Area?
0.8413
z
-3
-2
-1
0
1
2
3
Areas Under the Standard Normal
Curve

So, what is the area to the left of -1?
Area?
0.8413
z
-3
-2
-1
0
1
2
3
Areas Under the Standard Normal
Curve

So, what is the area between -1 and 1?
Area?
0.8413
0.8413
z
-3
-2
-1
0
1
2
3
Areas Under the Standard
Normal Curve

We will use two methods.
 The
TI-83 function normalcdf.
 Standard normal table.
TI-83 – Standard Normal Areas



Press 2nd DISTR.
Select normalcdf (Item #2).
Enter the lower and upper bounds of the
interval.
 If
the interval is infinite to the left, enter -E99 as the
lower bound.
 If the interval is infinite to the right, enter E99 as the
upper bound.

Press ENTER.
Standard Normal Areas

Use the TI-83 to find the following.
 The
area between -2 and +2.
 The area to the left of -2.
 The area to the right of -2.
Other Normal Curves

If we are working with a different normal
distribution, say N(30, 5), then how can we
find areas under the curve?
TI-83 – Area Under Normal Curves
Use the same procedure as before, except
enter the mean and standard deviation as
the 3rd and 4th parameters of the normalcdf
function.
 Find area between 25 and 38 in the
distribution N(30, 5).

IQ Scores




IQ scores are standardized to have a mean of
100 and a standard deviation of 15.
Psychologists often assume a normal
distribution of IQ scores as well.
What percentage of the population has an IQ
above 120? above 140?
What percentage of the population has an IQ
between 75 and 125?
The Standard Normal Table
See pages 406 – 407 or pages A-4 and A5 in Appendix A.
 The table is designed for the standard
normal distribution.
 The entries in the table are the areas to
the left of the z-value.

The Standard Normal Table

To find the area to the left of +1, locate
1.00 in the table and read the entry.
z
:
0.9
1.0
1.1
:
.00
:
0.8159
0.8413
0.8643
:
.01
:
0.8186
0.8438
0.8665
:
.02
:
0.8212
0.8461
0.8686
:
…
…
…
…
…
…
The Standard Normal Table

To find the area to the left of 2.31, locate
2.31 in the table and read the entry.
z
:
2.2
2.3
2.4
:
.00
:
0.9861
0.9893
0.9918
:
.01
:
0.9864
0.9896
0.9920
:
.02
:
0.9868
0.9898
0.9922
:
…
…
…
…
…
…
The Standard Normal Table
The area to the left of 1.00 is 0.8413.
 That means that 84.13% of the population
is below 1.00.

0.8413
-3
-2
-1
0
1
2
3
The Three Basic Problems

Find the area to the left of a:
 Look

up the value for a.
a
Find the area to the right of a:
 Look
up the value for a; subtract
it from 1.

Find the area between a and b:
a
 Look
up the values for a and b;
subtract the smaller value from
the larger.
a
b
Standard Normal Areas

Use the Standard Normal Tables to find
the following.
 The
area between -2.14 and +1.36.
 The area to the left of -1.42.
 The area to the right of -1.42.
Tables – Area Under Normal
Curves

If X is N(30, 5), what is the area to the left
of 35?
15
20
25
30
35
40
45
Tables – Area Under Normal
Curves

If X is N(30, 5), what is the area to the left
of 35?
15
20
25
30
35
40
45
Tables – Area Under Normal
Curves

If X is N(30, 5), what is the area to the left
of 35?
?
15
20
25
30
35
40
45
Tables – Area Under Normal
Curves

If X is N(30, 5), what is the area to the left
of 35?
?
X
15
20
25
30
35
40
45
-3
-2
-1
0
1
2
3
Z
Tables – Area Under Normal
Curves

If X is N(30, 5), what is the area to the left
of 35?
0.8413
X
15
20
25
30
35
40
45
-3
-2
-1
0
1
2
3
Z
Z-Scores
Z-score, or standard score
 Compute the z-score of x as
or
xx
x

z

z
s

Equivalently
x  x  zs
or
x    z
Areas Under Other Normal
Curves

If a variable X has a normal distribution,
then the z-scores of X have a standard
normal distribution.
If X is N (  ,  ), then
X 

is N (0,1)
Example


Let X be N(30, 5).
What proportion of values of X are below 38?
z = (38 – 30)/5 = 8/5 = 1.6.
 Find the area to the left of 1.6 under the standard
normal curve.
 Answer: 0.9452.
 Compute

Therefore, 94.52% of the values of X are below
38.
Bag A vs. Bag B
Suppose we have two bags, Bag A and
Bag B.
 Each bag contains millions of vouchers.
 In Bag A, the values of the vouchers have
distribution N(50, 10). In Bag B, the values
of the vouchers have distribution N(80,
15).

Bag A vs. Bag B
H0: Bag A
H1: Bag B
30
40
50
60
70
80
90
100
110
Bag A vs. Bag B

We select one voucher at random from
one bag.
H0: Bag A
H1: Bag B
30
40
50
60
70
80
90
100
110
Bag A vs. Bag B

If its value is less than or equal to $65,
then we will decide that it was from Bag A.
H0: Bag A
H1: Bag B
30
40
50
60
65
70
80
90
100
110
Bag A vs. Bag B

If its value is less than or equal to $65,
then we will decide that it was from Bag A.
H0: Bag A
H1: Bag B
30
40
50
Acceptance Region
60
65
70
80
90
100
110
Bag A vs. Bag B

If its value is less than or equal to $65,
then we will decide that it was from Bag A.
H0: Bag A
H1: Bag B
30
40
50
Acceptance Region
60
65
70
80
90
Rejection Region
100
110
Bag A vs. Bag B

What is ?
H0: Bag A
H1: Bag B
30
40
50
60
65
70
80
90
100
110
Bag A vs. Bag B

What is ?
H0: Bag A
H1: Bag B

30
40
50
60
65
70
80
90
100
110
Bag A vs. Bag B

What is ?
H0: Bag A
H1: Bag B
30
40
50
60
65
70
80
90
100
110
Bag A vs. Bag B

What is ?
H0: Bag A
H1: Bag B

30
40
50
60
65
70
80
90
100
110
Bag A vs. Bag B

If the distributions are very close together,
then  and  will be large.
H0: Bag A
H1: Bag B
N(60, 10)
30
40
50
N(70, 15)
60
65
70
80
90
100
110
Bag A vs. Bag B

If the distributions are very similar, then 
and  will be large.
H0: Bag A
H1: Bag B

30
40
50
60
65
70
80
90
100
110
Bag A vs. Bag B

If the distributions are very similar, then 
and  will be large.
H0: Bag A
H1: Bag B

30
40
50
60
65
70
80
90
100
110
Bag A vs. Bag B

Similarly, if the distributions are far apart,
then  and  will both be very small.
H0: Bag A
H1: Bag B
N(45, 10)
30
N(90, 15)
40
50
60
65
70
80
90
100
110
Bag A vs. Bag B

Similarly, if the distributions are far apart,
then  and  will both be very small.
H0: Bag A
H1: Bag B

30
40
50
60
65
70
80
90
100
110
Bag A vs. Bag B

Similarly, if the distributions are far apart,
then  and  will both be very small.
H0: Bag A
H1: Bag B

30
40
50
60
65
70
80
90
100
110