Transcript 6-2 - Fenwick High School

Normal Curves and
Sampling
Distributions
6
Copyright © Cengage Learning. All rights reserved.
Section
6.2
Standard Units and
Areas Under the
Standard Normal
Distribution
Copyright © Cengage Learning. All rights reserved.
Focus Points
•
Given  and , convert raw data to z scores.
•
Given  and , convert z scores to raw data.
•
Graph the standard normal distribution, and find
areas under the standard normal curve.
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z Scores and Raw Scores
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z Scores and Raw Scores
Normal distributions vary from one another in two ways:
The mean  may be located anywhere on the x axis, and
the bell shape may be more or less spread according to the
size of the standard deviation .
The differences among the normal distributions cause
difficulties when we try to compute the area under the curve
in a specified interval of x values and, hence, the
probability that a measurement will fall into that interval.
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z Scores and Raw Scores
It would be a futile task to try to set up a table of areas
under the normal curve for each different  and 
combination. We need a way to standardize the
distributions so that we can use one table of areas for all
normal distributions.
We achieve this standardization by considering how many
standard deviations a measurement lies from the mean. In
this way, we can compare a value in one normal
distribution with a value in another, different normal
distribution. The next situation shows how this is done.
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z Scores and Raw Scores
Suppose Tina and Jack are in two different sections of the
same course. Each section is quite large, and the scores
on the midterm exams of each section follow a normal
distribution. In Tina’s section, the average (mean) was 64
and her score was 74. In Jack’s section, the mean was 72
and his score was 82.
Both Tina and Jack were pleased that their scores were
each 10 points above the average of each respective
section. However, the fact that each was 10 points above
average does not really tell us how each did with respect to
the other students in the section.
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z Scores and Raw Scores
In Figure 6-14, we see the normal distribution of grades for
each section.
Distributions of Midterm Scores
Figure 6-14
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z Scores and Raw Scores
Tina’s 74 was higher than most of the other scores in her
section, while Jack’s 82 is only an upper-middle score in
his section. Tina’s score is far better with respect to her
class than Jack’s score is with respect to his class.
The preceding situation demonstrates that it is not sufficient
to know the difference between a measurement (x value)
and the mean of a distribution. We need also to consider
the spread of the curve, or the standard deviation.
What we really want to know is the number of standard
deviations between a measurement and the mean. This
“distance” takes both  and  into account.
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z Scores and Raw Scores
We can use a simple formula to compute the number z of
standard deviations between a measurement x and the
mean  of a normal distribution with standard deviation :
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z Scores and Raw Scores
The mean is a special value of a distribution. Let’s see
what happens when we convert x =  to a z value:
The mean of the original distribution is always zero, in
standard units. This makes sense because the mean is
zero standard variations from itself.
An x value in the original distribution that is above the
mean  has a corresponding z value that is positive.
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z Scores and Raw Scores
Again, this makes sense because a measurement above
the mean would be a positive number of standard
deviations from the mean. Likewise, an x value below the
mean has a negative z value. (See Table 6-2.)
x Values and Corresponding z Values
Table 6-2
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Example 4 – Standard score
A pizza parlor franchise specifies that the average (mean)
amount of cheese on a large pizza should be 8 ounces and
the standard deviation only 0.5 ounce. An inspector picks
out a large pizza at random in one of the pizza parlors and
finds that it is made with 6.9 ounces of cheese.
Assume that the amount of cheese on a pizza follows a
normal distribution. If the amount of cheese is below the
mean by more than 3 standard deviations, the parlor will be
in danger of losing its franchise.
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Example 4 – Standard score
How many standard deviations from the mean is 6.9? Is the
pizza parlor in danger of losing its franchise?
Solution:
Since we want to know the number of standard deviations
from the mean, we want to convert 6.9 to standard z units.
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z Scores and Raw Scores
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Standard Normal Distribution
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Standard Normal Distribution
If the original distribution of x values is normal, then the
corresponding z values have a normal distribution as well.
The z distribution has a mean of 0 and a standard deviation
of 1. The normal curve with these properties has a special
name.
Any normal distribution of x values can be converted to the
standard normal distribution by converting all x values to
their corresponding z values. The resulting standard
distribution will always have mean  = 0 and standard
deviation  = 1.
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Areas Under the Standard Normal Curve
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Areas Under the Standard Normal Curve
We have seen how to convert any normal distribution to the
standard normal distribution. We can change any x value to
a z value and back again. But what is the advantage of all
this work?
The advantage is that there are extensive tables that show
the area under the standard normal curve for almost any
interval along the z axis.
The areas are important because each area is equal to the
probability that the measurement of an item selected at
random falls in this interval.
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Areas Under the Standard Normal Curve
Thus, the standard normal distribution can be a
tremendously helpful tool.
Addition of 3 and 4
The Standard Normal Distribution ( = 0,  = 1)
FIGURE 6-15
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Using a Standard Normal Distribution Table
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Using a Standard Normal Distribution Table
Using a table to find areas and probabilities associated with
the standard normal distribution is a fairly straightforward
activity.
However, it is important to first observe the range of z
values for which areas are given. This range is usually
depicted in a picture that accompanies the table.
In this text, we will use the left-tail style table. This style
table gives cumulative areas to the left of a specified z.
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Using a Standard Normal Distribution Table
Determining other areas under the curve utilizes the fact
that the area under the entire curve is 1.
Taking advantage of the symmetry of the normal
distribution is also useful.
The procedures you learn for using the left-tail style normal
distribution table apply directly to cumulative normal
distribution areas found on calculators and in computer
software packages such as Excel 2007 and Minitab.
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Example 5(a) – Standard normal distribution table
Use Table 5 of Appendix II to find the described areas
under the standard normal curve.
(a) Find the area under the standard normal curve to the
left of z = –1.00.
Solution:
First, shade the area to be found
on the standard normal distribution
curve, as shown in Figure 6-16.
Area to the Left of z = –1.00
Figure 6-16
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Example 5(a) – Solution
cont’d
Notice that the z value we are using is negative. This
means that we will look at the portion of Table 5 of
Appendix II for which the z values are negative. In the
upper-left corner of the table we see the letter z.
The column under z gives us the units value and tenths
value for z. The other column headings indicate the
hundredths value of z. Table entries give areas under the
standard normal curve to the left of the listed z values.
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Example 5(a) – Solution
cont’d
To find the area to the left of z = –1.00, we use the row
headed by –1.0 and then move to the column headed by
the hundredths position, .00. This entry is shaded in Table
6-3. We see that the area is 0.1587.
Excerpt from Table 5 of Appendix II Showing Negative z Values
Table 6-3
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Example 5(b) – Standard normal distribution table
cont’d
(b) Find the area to the left of z = 1.18, as illustrated in
Figure 6-17.
Area to the Left of z = 1.18
Figure 6-17
Solution:
In this case, we are looking for an area to the left of a
positive z value, so we look in the portion of Table 5 that
shows positive z values.
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Example 5(b) – Solution
cont’d
Again, we first sketch the area to be found on a standard
normal curve, as shown in Figure 6-17.
Excerpt from Table 5 of Appendix II Showing Positive z Values
Table 6-4
Look in the row headed by 1.1 and move to the column
headed by .08. The desired area is shaded (see Table 6-4).
We see that the area to the left of 1.18 is 0.8810.
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Using a Standard Normal Distribution Table
Procedure:
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Using a Standard Normal Distribution Table
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Example 6(a) – Using table to find areas
Use Table 5 of Appendix II to find the specified areas.
Find the area between z = 1.00 and z = 2.70.
Solution:
First, sketch a diagram showing the area (see Figure 6-19).
Area from z = 1.00 to z = 2.70
Figure 6-19
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Example 6(a) – Solution
cont’d
Because we are finding the area between two z values, we
subtract corresponding table entries.
(Area between 1.00 and 2.70) = (Area left of 2.70)
– (Area left of 1.00)
= 0.9965 – 08413
= 0.1552
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Example 6(b) – Using table to find areas cont’d
Find the area to the right of z = 0.94.
Solution:
First, sketch the area to be found (see Figure 6-20).
Area to the Right of z = 0.94.
Figure 6-20
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Example 6(b) – Solution
cont’d
(Area to right of 0.94) = (Area under entire curve)
– (Area to left of 0.94)
= 1.000 – 0.8264
= 0.1736
Alternatively,
(Area to right of 0.94) = (Area to left of – 0.94)
= 0.1736
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