Download BEG_6_2

Section 6.2
Finding Area under a Normal
Distribution
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Objectives
o Find areas under the standard normal distribution.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.2: Finding Area to the Left of a Positive
z-Value Using a Cumulative Normal Table
Find the area under the standard normal curve to the
left of z = 1.37.
Solution
To read the table, we must break the given z-value
(1.37) into two parts: one containing the first decimal
place (1.3) and the other containing the second
decimal place (0.07). So, in Table B from Appendix A,
look across the row labeled 1.3 and down the column
labeled 0.07. The row and column intersect at 0.9147.
Thus, the area under the standard normal curve to the
left of z = 1.37 is 0.9147.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.2: Finding Area to the Left of a Positive
z-Value Using a Cumulative Normal Table (cont.)
z
1.0
1.1
1.2
1.3
1.4
1.5
0.05
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
0.06
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.07
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.08
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.09
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.2: Finding Area to the Left of a Positive
z-Value Using a Cumulative Normal Table (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.3: Finding Area to the Left of a Negative
z-Value Using a Table or a TI-83/84 Plus Calculator
Find the area under the standard normal curve to the
left of z = −2.03.
Solution
The first part of the z-value is -2.0 and the second part
is 0.03. This time, use Table A from Appendix A since
the z-value is negative; look across the row labeled -2.0
and down the column labeled 0.03. The row and
column intersect at 0.0212. Thus, the area under the
normal curve to the left of z = −2.03 is 0.0212.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.3: Finding Area to the Left of a Negative zValue
Using a Table or a TI-83/84 Plus Calculator (cont.)
z
-2.2
-2.1
-2.0
-1.9
-1.8
-1.7
0.04
0.0125
0.0162
0.0207
0.0262
0.0329
0.0409
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
0.03
0.0129
0.0166
0.0212
0.0268
0.0336
0.0418
0.02
0.0132
0.0170
0.0217
0.0274
0.0344
0.0427
0.01
0.0136
0.0174
0.0222
0.0281
0.0351
0.0436
0
0.0139
0.0179
0.0228
0.0287
0.0359
0.0446
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.3: Finding Area to the Left of a Negative zValue
Using a Table or a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.3: Finding Area to the Left of a Negative zValue
Using a Table or a TI-83/84 Plus Calculator (cont.)
To obtain the solution using a TI-83/84 Plus calculator,
perform the following steps.
• Press
and then
to access the DISTR
menu.
• Choose option 2:normalcdf(.
• Enter lower bound, upper bound, m, s. Note: If
you want to find area under the standard normal
curve, as in this example, then you do not need
to enter m or s.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.3: Finding Area to the Left of a Negative zValue
Using a Table or a TI-83/84 Plus Calculator (cont.)
• Since we are asked to find the area to the left of
z, the lower bound is -∞. We cannot enter -∞
into the calculator, so we will enter a very small
value for the lower endpoint, such as -1099. This
number appears as -1E99 when entered
correctly into the calculator. To enter -1E99,
press
.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.3: Finding Area to the Left of a Negative zValue
Using a Table or a TI-83/84 Plus Calculator (cont.)
Enter normalcdf(-1E99,-2.03), as shown in the
screen shot. The area is approximately 0.0212.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.4 : Finding Area to the Right of a Positive
z-Value Using a Cumulative Normal Table
Find the area under the standard normal curve to the
right of z = 1.37.
Solution
• Method 1: From Example 6.2, we know that the
area under the standard normal curve to the left
of z = 1.37 is 0.9147. So, the area under the
standard normal curve to the right of z = 1.37 is
1 − 0.9147 = 0.0853.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.4 : Finding Area to the Right of a Positive
z-Value Using a Cumulative Normal Table (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.4 : Finding Area to the Right of a Positive
z-Value Using a Cumulative Normal Table (cont.)
• Method 2: We can look up z = −1.37 in Table A
from Appendix A, which also gives us 0.0853.
z
-1.6
-1.5
-1.4
-1.3
-1.2
-1.1
0.09
0.0455
0.0559
0.0681
0.0823
0.0985
0.1170
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
0.08
0.0465
0.0571
0.0694
0.0838
0.1003
0.1190
0.07
0.0475
0.0582
0.0708
0.0853
0.1020
0.1210
0.06
0.0485
0.0594
0.0721
0.0869
0.1038
0.1230
0.05
0.0495
0.0606
0.0735
0.0885
0.1056
0.1251
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.5: Finding Area to the Right of a Negative
z-Value Using a Table or a TI-83/84 Plus Calculator
Find the area under the standard normal curve to the
right of z = -0.90.
Solution
If we look up z = -0.90 in Table A from Appendix A, we
see that an area of 0.1841 lies to the left of z. Then,
subtract this area from 1 to find the amount of area to
the right of z. Hence, an area of 1 - 0.1841 = 0.8159
lies to the right of z. Note that this is the same area you
find by simply looking up z = 0.90 in Table B from
Appendix A.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.5: Finding Area to the Right of a Negative zValue
Using a Table or a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.5: Finding Area to the Right of a Negative zValue
Using a Table or a TI-83/84 Plus Calculator (cont.)
To obtain the solution using a TI-83/84 Plus calculator,
perform the following steps.
• Press
and then
to access the DISTR
menu.
• Choose option 2:normalcdf(.
• Because the calculator will not allow us to enter
∞ for the upper endpoint, we will choose a
sufficiently large number, such as 1099. This
number appears as 1û99 when entered
correctly into the calculator. To enter 1û99 into
the calculator, press
.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.5: Finding Area to the Right of a Negative zValue
Using a Table or a TI-83/84 Plus Calculator (cont.)
Thus, to obtain the answer, we enter
normalcdf(-0.90,1û99) into the calculator, as
shown in the screenshot, and find that the area to the
right of z = -0.90 is indeed approximately 0.8159.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.6: Finding Area between Two z-Values
Using Tables or a TI-83/84 Plus Calculator
Find the area under the standard normal curve
between z1 = -1.68 and z2 = 2.00.
Solution
First, look up the area to the left of z1 = -1.68, which is
0.0465. Second, look up the area to the left of z2 = 2.00,
which is 0.9772. Finally, subtract: 0.9772 - 0.0465 =
0.9307. Thus, the area between the two z-values is
0.9307.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.6: Finding Area between Two z-Values
Using Tables or a TI-83/84 Plus Calculator (cont.)
Enter normalcdf(-1.68,2.00), as shown in the
screenshot on next slide, to find that the area between
z1 = -1.68 and z2 = 2.00 is approximately 0.9308.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.6: Finding Area between Two z-Values
Using Tables or a TI-83/84 Plus Calculator (cont.)
Notice that this is a slightly different answer than what
we obtained using the tables. The reason is that the
table values have been rounded in the intermediate
steps and the calculator values have not. For this
reason, calculator solutions may vary slightly from
those obtained by using the tables.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.7: Finding Area between Two
z-Values Using a TI-83/84 Plus Calculator
Find the area under the standard normal curve
between z1 = 1.50 and z2 = 2.75.
Solution
Let’s use a TI-83/84 Plus calculator to find this area. We
want to use a lower bound of 1.50 and an upper bound
of 2.75. Enter normalcdf(1.50,2.75) into the
calculator, which gives a value of approximately 0.0638
as shown in the screenshot on next slide.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.7: Finding Area between Two
z-Values Using a TI-83/84 Plus Calculator (cont.)
Therefore, we see that an area of approximately 0.0638
lies between z1 = 1.50 and z2 = 2.75.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.7: Finding Area between Two
z-Values Using a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.8: Finding Area in the Tails for Two
z-Values Using a TI-83/84 Plus Calculator
Find the total of the areas under the standard normal
curve to the left of z1 = −2.50 and to the right of
z2 = 3.00.
Solution
There are two areas that we must find. The total we are
interested in is the sum of these two areas. Let’s begin
by finding the area to the left of z1 = −2.50. Enter
normalcdf(-1û99,-2.50) to find that the area to
the left of z1 = −2.50 is approximately 0.006210, as
shown in the screenshot on next slide.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.8: Finding Area in the Tails for Two
z-Values Using a TI-83/84 Plus Calculator (cont.)
Next, we need to find the area to the right of z2 = 3.00.
Enter normalcdf(3.00,1û99) to find that the
area to the right of z2 = 3.00 is approximately 0.001350,
as shown in the screenshot. Thus, the total area in the
tails is the sum of the two areas, 0.006210 + 0.001350
≈ 0.0076.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.8: Finding Area in the Tails for Two
z-Values Using a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.8: Finding Area in the Tails for Two
z-Values Using a TI-83/84 Plus Calculator (cont.)
Note an alternative method for finding this area that is
particularly clever. By definition, we know that the total
area under the curve equals 1. Using this fact, the area
in the tails can be obtained by finding the area between
z1 = −2.50 and z2 = 3.00 and then subtracting that area
from 1. This method can even be entered in one step
into your calculator as shown below and in the
screenshot on next slide.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.8: Finding Area in the Tails for Two
z-Values Using a TI-83/84 Plus Calculator (cont.)
1Þnormalcdf(-2.50,3.00) ≈ 0.0076
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.9: Finding Area in the Tails for Two
z-Values Using a TI-83/84 Plus Calculator
Find the total of the areas under the standard normal
curve to the left of z1 = -1.23 and to the right of
z2 = 1.23.
Solution
Notice that the absolute values of z1 and z2 are the
same. Thus, the areas in the two tails of the
distribution will be the same because of the symmetric
property of the standard normal curve. So, to find the
total area in the two tails, we only need to look up the
area in the left tail and multiply that by 2.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.9: Finding Area in the Tails for Two
z-Values Using a TI-83/84 Plus Calculator (cont.)
By entering normalcdf(-1û99,-1.23), we find
that the area to the left of z1 = -1.23 is approximately
0.109349, as shown in the screenshot. Multiply this
area by 2 in order to obtain the combined area in the
tails.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.9: Finding Area in the Tails for Two
z-Values Using a TI-83/84 Plus Calculator (cont.)
Thus, (0.109349)(2)  0.2187. So, the total area in the
two tails is approximately 0.2187.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Finding Area under a Normal Distribution
Using the Cumulative Normal Distribution Tables to
Find Areas under the Standard Normal Curve
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Finding Area under a Normal Distribution
Using the Cumulative Normal Distribution Tables to
Find Areas under the Standard Normal Curve (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Finding Area under a Normal Distribution
Using a TI-83/84 Plus Calculator to Find Areas under
the Standard Normal Curve
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Finding Area under a Normal Distribution
Using a TI-83/84 Plus Calculator to Find Areas under
the Standard Normal Curve (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.10: Interpreting Probability for the Standard
Normal Distribution as an Area under the Curve
Interpret P(z  −2.67).
Solution
P(z  −2.67) stands for the probability that z is less than
or equal to -2.67. This is equal to the area under the
standard normal curve to the left of z = −2.67.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.10: Interpreting Probability for the Standard
Normal Distribution as an Area under the Curve (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.10: Interpreting Probability for the Standard
Normal Distribution as an Area under the Curve (cont.)
Note that the probability that z is less than a value is
the same as the probability that z is less than or equal
to that value, or symbolically,
P(z  −2.67) = P(z < −2.67).
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator
Find the following probabilities using the cumulative
normal distribution tables or a TI-83/84 Plus calculator.
a. P(z < 1.45)
b. P(z  −1.37)
c. P(1.25 < z < 2.31)
d. P(z < −2.5 or z > 2.5)
e. P(z < −4.01)
f. P(z  3.98)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
Solution
For each part of this example, the solution first
describes the method for finding the answer using the
cumulative normal distribution tables and then explains
how to find the answer using a TI-83/84 Plus calculator.
a. P(z < 1.45) is the area under the standard normal
curve to the left of z = 1.45. Look up z = 1.45 in the
cumulative normal table for positive z-values,
Table B. The area is 0.9265.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
By entering normalcdf(-1û99,1.45), we find
that the area to the left of z = 1.45 is approximately
0.9265, as shown in the screenshot.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
b. P(z  −1.37) is the area under the standard normal
curve to the right of z = −1.37. Use the symmetry
property of the standard normal curve and look up
z = 1.37 in the cumulative normal table for positive
z-values, Table B. The area to the right of z = −1.37 is
0.9147.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
By entering normalcdf(-1.37,1û99), we find
that the area to the right of z = −1.37 is approximately
0.9147, as shown in the screenshot.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
c. P(1.25 < z < 2.31) is the area under the standard
normal curve between z1 = 1.25 and z2 = 2.31. Look
up each value in the cumulative normal table for
positive z-values, Table B. The area to the left of
z1 = 1.25 is 0.8944. The area to the left of z2 = 2.31 is
0.9896. The area between z1 = 1.25 and z2 = 2.31 is
the difference between these two areas. Thus, the
area is 0.9896 − 0.8944 = 0.0952.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
By entering normalcdf(1.25,2.31), we find that
the area between z1 = 1.25 and z2 = 2.31 is
approximately 0.0952, as shown in the screenshot.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
d. P(z < −2.5 or z > 2.5) is the sum of the areas to the
left of z1 = −2.5 and to the right of z2 = 2.5. Since the
normal curve is symmetric, these two areas are the
same; therefore, look up the area to the left of
z = −2.50 in the cumulative normal table for
negative z-values, Table A, and multiply that area by
2. The area to the left of z = −2.50 is 0.0062. Thus,
the area we are interested in is (0.0062)(2) = 0.0124.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
Enter 1Þnormalcdf(-2.5,2.5). We find that the
sum of the areas to the left of z1 = −2.5 and to the right
of z2 = 2.5 is approximately 0.0124, as shown in the
screenshot.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
e. P(z < −4.01) is the area under the standard normal
curve to the left of z = −4.01. Notice that z = −4.01 is
not in the cumulative normal table for negative
z-values, Table A. It is smaller than the z-values in
the table, which means that it is further to the left.
Thus, the area under the curve is smaller than all of
the areas listed in the table. Therefore, P(z < −4.01)
is approximately 0.0000.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
By entering normalcdf(-1û99,-4.01), we find
that the area to the left of z = −4.01 is approximately
0.00003, as shown in the screenshot.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
f. P(z  3.98) is the area under the standard normal
curve to the left of z = 3.98. Notice that z = 3.98 is
not in the cumulative normal table for positive
z-values, Table B. It is larger than the z-values in the
table, which means that it is further to the right.
Thus, the area under the curve is larger than all of
the areas listed in the table. Therefore, P(z  3.98) is
approximately 1.0000.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6.11 : Finding Probabilities for the Standard Normal
Distribution Using Tables or a TI-83/84 Plus Calculator (cont.)
By entering normalcdf(-1û99,3.98),we find
that the area to the left of z = 3.98 is approximately
1.0000, as shown in the screenshot.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.