CHAPTER 2: THE NORMAL
DISTRIBUTIONS
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:
 We want to find the SAT score x with area .1 to its
right, which means .9 will be to its left
 Use the table and draw a sketch:
 .9 on the table corresponds best to z = 1.28
x  505
z0
x?
z  1.28
EXAMPLE 1: SAT VERBAL SCORES

Unstandardize to transform the solution from z
back to the original x scale:
x  505
 1.28
110
x  505  140.8

 x  505 
110 
  1.28  110
 110 
x  645.8
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.
QUICK CHECK

Do #24c on p.103
ASSESSING NORMALITY

Since we will eventually be using various tests of
significance that require the normal distibution
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 ( x ) 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
ASSESSING NORMALITY

Method 2: Construct a normal probability plot
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
ASSESSING NORMALITY

Method 2: Construct a boxplot using your graphing
calculator
Enter the values in L1
 First go to STAT/CALC/1-VarStats to compare the mean and
median



To construct a boxplot:


If the values are approximately normal, then the mean and median
should be close to one another
Go to 2nd/Y=, turn the plot ON, choose the 4th graph (boxplot with
outliers), and then choose Zoom9
 Look for symmetry
 Make sure the whiskers extend far enough in comparison to the box
 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.


Data set A:
 Normal probability plot:
 there is a nonlinear trend, so it does not take on a normal shape
 Boxplot:
 It is right skewed and therefore does not take on a normal shape
Data set B:
 Normal probability plot:
 Aside from one outlier, it takes on a linear trend and is therefore
approximately normal
 Boxplot:
 Aside from one outlier, the boxplot appears to be fairly symmetric
and is therefore considered approximately normal
FINDING AREAS WITH NORMALCDF
 The
normalcdf command on the TI-83/84
can be used to find the area under a
normal distribution and above an interval
 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?
normalcdf(-1000,400,505,110)


About 17% of all tests will have a score lower than 400
What is the proportion of scores that are above 600?
normalcdf(600,2000,505,110)


About 19% of all tests will have a score higher than 600
What is the proportion of scores that are between 480 and
650?
normalcdf(480,650,505,110)

About 50% of all tests will have a score between 480 and 650
QUICK CHECK

Redo #23a & b on p.103 using calc
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
invNorm(.85,505,110)

A student must score a 619 or higher to be in the top 15% of
all students on the SAT Verbal test.
QUICK CHECK

#43 on p.113 using calc
SECTION 2.2 DAY 2 HOMEWORK

P.103-114 #’s 22, 23c, 25, 27, 30, 31c, 33, & 47