Section 2.2 ~ Standard Normal Calculations
Finding a value given a proportion:

If you want to find the observed value that pertains to a given proportion:

Use the standard normal table backwards to “unstandardize”

Find the given proportion in the body of the table and determine the z-score that corresponds

Multiply that z-score by the value of 1 St.Dev. and then add or subtract it from the value of the mean
Example 1 – SAT Verbal Scores
Scores on the SAT Verbal test in recent years follow approximately the N(505, 110) distribution. How high must a
student score in order to place in the top 10% of all students taking the SAT?

State the problem:


Use the table and draw a sketch:


We want to find the SAT score x with area .1 to its right, which means .9 will be to its left
.9 on the table corresponds best to z = 1.28
Unstandardize to transform the solution from z back to the original x scale:


Write in context:

A person with a score of 645.8 or higher on the SAT Verbal test would be in the top 10% of all students
taking the test.
Assessing Normality:
Since we will eventually be using various tests of significance that require the normal distribution to gain insights in
regards to a population, it is important to develop methods for assessing normality.

Method 1: Construct a frequency histogram or stemplot using your sample observations
 See if the graph is approximately bell shaped and symmetric about the mean

Do this by finding the mean ( ) and St.Dev. (s) of the sample observations and compare the
count of observations to the 68-95-99.7 rule
 See example 2.11 on p.104

Method 2: Construct a normal probability plot or a boxplot using your graphing calculator
 Enter the values in L1
 First go to STAT/CALC/1-VarStats to compare the mean and median
 If the values are approximately normal, then the mean and median should be close to one
another



To construct a normal probability plot:
 Go to 2nd/Y=, turn the plot ON, choose the last type of graph (normal probability plot), and then
choose Zoom9 (ZoomStat)
 If the data is close to normal, the plotted points will lie close to a straight line

If the

data are nonnormal, there will be a nonlinear trend
To construct a boxplot:
 Enter the values in L1
 First go to STAT/CALC/1-VarStats to compare the mean and median
 If the values are approximately normal, then the mean and median should be close to
one another
nd
 Go to 2 /Y=, turn the plot ON, choose the 4th graph (boxplot with outliers), and then choose
Zoom9
 Look

for symmetry

On the AP exam you must show that you verified the assumption of normality and since
boxplots are much easier to display than normal probability plots, this is typically the
method of choice
Method 3: Make a graph and look at it
 While this method is the least mathematical, it is still acceptable to just make a graph and look at it to
determine if it appears to be approximately normal
Example 2 – Assessing Normality
Use the TI-83/84 to construct a normal probability plot as well as a boxplot for the following sets of data, and use the
plots to assess the normality of the data.
Finding areas with normalcdf:

The normalcdf command on the TI-83/84 can be used to find the areas under a normal distribution

Press 2nd/VARS/normalcdf(

Enter the (lower bound, upper bound, mean, st.dev.)

If you want the area to the left: then pick an extremely low value to represent the lower
bound (which is theoretically negative infinity)

If you want the area to the right: then pick an extremely high value to represent the upper
bound (which is theoretically positive infinity)
Example 3 – Using normalcdf
Scores on the SAT Verbal test in recent years follow approximately the N(505, 110) distribution.

What is the proportion of scores that are below 400?


What is the proportion of scores that are above 600?


About 17% of all tests will have a score lower than 400
About 19% of all tests will have a score higher than 600
What is the proportion of scores that are between 480 and 650?

About 50% of all tests will have a score between 480 and 650
Finding z-scores with invNorm:

The invNorm function calculates the raw or standardized normal value corresponding to a known area under a
normal distribution

Press 2nd/VARS/invNorm(

For the actual raw value: Enter the (area, mean, st.dev.)

For the standardized value (z-score): Enter invNorm(area)
Example 4 – Using invNorm
Scores on the SAT Verbal test in recent years follow approximately the N(505, 110) distribution. How high must a
student score in order to place in the top 15% of all students taking the SAT?

The top 15% corresponds to the 85th percentile

A student must score a 619 or higher to be in the top 15% of all students on the SAT Verbal test.