2. Lesson Objectives
At the end of this lesson, you are expected to:
1. identify the regions of the areas under the
normal curve;
2. express the areas under the normal curve as
probabilities or percentages; and
3. determine the areas under the normal curve
given z-values.
4. Lesson Introduction
The area under the curve is 1. So, we can
make the correspondence between area
and probability.
5. Discussion Points
Identifying Regions Under the Normal Curve
z-table provides the proportion of the area (or probability or
percentage) between any two specific values under the curve,
regions under the curve can be described in terms of area.
For example, the area of the region between z = 0 and z = 1 is given
in the z-table to be .3413.
6. Discussion Points
To find the area of the region between z = 1
and z = 2, we subtract .3413 from .4772
resulting in .1359. It is graphically shown
below.
7. Discussion Points
The regions under the normal curve in terms
of percent, the graph of the distribution
would look like this:
8. Discussion Points
Using the z-Table in Determining Areas Under
the Normal Curve when z is Given
Step 1. Write the given z-value into a three-digit
form.
Step 2.Find the first two digits in row.
Step 3. Locate the third digit in Column
Step 4. Take the area value at the intersection of
Row and Column.
11. Exercises
Use the z-table to find the area that
corresponds between z=-1 to each of the
following:
1. z=0.56
2. z= 1.32
3. z = –1.05
4. z = –2.18
5. z = –2.58
12. Do the indicated task.
1. Explain why the proportion of the area to the
left of z = –2.58 is .49%.
2. Explain why the total area of the region
between z = –3 and z = 3 is 9974 or 99.74%.
13. Summary
Properties of the Normal Probability Distribution
The distribution curve is bell-shaped.
The curve is symmetrical about its center.
The mean, the median, and the mode coincide at the center.
The width of the curve is determined by the standard
deviation of the distribution.
The tails of the curve flatten out indefinitely along the
horizontal axis, always approaching the axis but never
touching it. That is, the curve is asymptotic to the base line.
The area under the curve is 1. Thus, it represents the
probability or proportion or the percentage associated with
specific sets of measurement values.
14. Summary
Using the z-Table in Determining Areas Under
the Normal Curve when z is Given
Step 1. Write the given z-value into a three-digit
form.
Step 2.Find the first two digits in row.
Step 3. Locate the third digit in Column
Step 4. Take the area value at the intersection of
Row and Column.
