Download 7.2.1 Finding Areas under the Standard Normal Curve

The Standard Normal Distribution
Page 1
The Standard Normal Distribution
The goal of this activity is to use various methods to find the area under the standard normal
curve.
Using Table A
Table A from that AP Statistics Exam provides the
cumulative distribution function for values of z rounded to
the nearest hundredth between -3.49 and 3.49. This table
provides the area under the standard normal curve for values
of z less than those identified in the table. This is illustrated
in the figure on the right (and at the top of the AP Table
page) with the shaded region, labeled probability.
The figure below demonstrates how to use the table to find the area under the standard normal
curve that lies to the left of Z = 1.46. Notice that the value 1.46 = 1.4 + .06. The value 1.4 is
found by scrolling down the first column of the table and the value .06 is found by moving right
across the top row. The intersection within the table of the row of 1.4 and the column of .06 is
the value .9279. This is the area under the normal curve to the left of Z = 1.46.
Use Table A to find the following areas under the standard normal curve.
1. The area that lies to the left of Z = -0.58.
2. The area that lies between Z = -1.16 and Z = 2.71.
3. The area that lies to the right of Z = 0.31.
Robert A. Powers
University of Northern Colorado
The Standard Normal Distribution
Page 2
Using Technology
The TI graphing calculator provides a built in function for determining the cumulative
distribution function. It is located in the DISTR menu above the VARS key. The syntax of the
function is given below.
normalcdf(lower,upper,µ,σ)
For example, to find the area under the standard normal curve that lies to the left of Z = 1.46, the
following command would be entered.
normalcdf(ú1E99,1.46,0,1)
Notice that ú1E99 is used to represent -∞.
1. Find the area under the standard normal curve that lies to the left of Z = -1.31.
One advantage of using technology is that finding the area under the normal curve between two
values does not require multiple steps. For example, when finding the area between Z = -0.61
and Z = 1.54, simple enter -.61 as the lower bound and 1.54 as the upper bound.
normalcdf(ú.61,1.54,0,1)
2. Find the area under the standard normal curve that lies between Z = 1.07 and Z = 3.29.
3. Find the area under the standard normal curve that lies to the right of Z = -2.40.
Another advantage of using technology is that finding the area under the normal curve is not
restricted to values that are rounded to the nearest hundredth.
4. Find the area under the standard normal curve that lies to the left of Z = 1.645.
Finally, an advantage of using technology is that finding the area under a normal curve is not
restricted to the standard normal curve.
5. To demonstrate this, find the area under the normal curve with µ = 266 and σ = 16 within one
standard deviation of the mean.
Robert A. Powers
University of Northern Colorado
The Standard Normal Distribution
Page 3
The Standard Normal Distribution
Often times, we are interested in finding the Z-score that corresponds to a given area under the
standard normal curve. The process involves searching the array of area values and working
backwards to find the Z-score.
Using Table A
Example: Find the Z-score that corresponds to an area of 0.9050 under the standard normal curve
to the left of the Z-score.
When searching the array of values, the closes one we see is .9049. This value is in the row of
1.3 and the column of .01. Thus, the Z-score is 1.31.
1. Find the Z-score so that the area to the left of the Z-score is 0.10.
2. Find the Z-score so that the area to the right of the Z-score is 0.0735.
Using Technology
The TI graphing calculator provides a built in function for determining the Z-score from the
cumulative distribution function. The syntax of the function is given below.
invNorm(area,µ,σ)
Example: Find the Z-score that corresponds to an area of 0.9050 under the standard normal curve
to the left of the Z-score using the TI calculator. Enter the following:
invNorm(.9050,0,1)
Robert A. Powers
University of Northern Colorado
The Standard Normal Distribution
Page 4
3. Find the Z-score so that the area to the left of the Z-score is 0.25.
4. Find the Z-score so that the area to the right of the Z-score is 0.3333.
We are often interested in finding the Z-score that has a specified area to the right. For this
reason, we have special notation to represent this situation.
The notation zα (pronounced “Z sub alpha”) is the Z-score such that the area under the
standard normal curve to the right of zα is α.
5. Find the value of z0.05.
Recall that the area under the standard normal distribution can be interpreted as either a
probability or as the proportion of the population with the given characteristic. When
interpreting the area under the standard normal curve as a probability, we use the following
notation.
Notation for the Probability of a Standard Normal Random Variable
P(a < Z < b)
P(Z > a)
P(Z < a)
represents the probability that a standard normal random variable is between a
and b.
represents the probability that a standard normal random variable is greater
than a.
represents the probability that a standard normal random variable is less than a.
6. Evaluate P(Z < 2.73)
7. Evaluate P(Z > 0.84)
Robert A. Powers
University of Northern Colorado