Download Solving Problems Involving Normal Distributions

AP Statistics
2.2 Normal Distribution Calculations
Objectives: *


Identify at least two graphical techniques for assessing Normality.
Explain what is meant by a Normal probability plot, use it to help assess the
Normality of a given data set.
Use technology to perform Normal distribution calculations and to make Normal
probability plots.
Here is an outline of the method for finding the proportion of the distribution in any region:
Solving Problems Involving Normal Distributions:
Step 1:
Step 2:
Step 3:
Step 4:
State the problem. Write down all the info that you are given. Sketch a picture,
shade the desired area, and label the mean.
Standardize the observation, sketch a picture, shade the desired area, and label the
mean.
Use Table A or your calculator to find the proportion of observations in your interval.
Write your conclusion and remember that it needs to be in the CONTEXT OF THE
PROBLEM!
An example is done for you in your book on page 143. It is a good idea to study it.
Example (pg.146; 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? (Finding a proportion given a certain value)
Step 1 (state the problem): We want to find the SAT score x with area 0.1 to its right under the
Normal curve with mean 505 and standard deviation 110. That’s the same as find the SAT score x
with are 0.9 to its left. Figure 2.20 poses the question in graphical form. DRAW IT BELOW.
Step 2 (use the table): Look in Table A for the entry closest to 0.9, it is ______________. This is the entry
corresponding to z = _________________, so that is the standardized value with area 0.9 to the left.
Step 3 (unstandardized): transform the solution from the z scale back to the original x scale.
Step 4 (conclusion): We see that a student must score at least __________ to place in the highest 10%.
Exercise 2.31 (page 147)
Assessing Normality: How do I know if a distribution is Normal?
METHOD 1: Construct a histogram or a stem plot
 This is not enough:
1. Label x̅ , x̅ ± s, x̅ ± 2s, x̅ ± 3s
2. Compare the proportion in each to the 68 – 95 – 99.7 rule. If the proportions in each
are close to these percents, it is safe to say it is Normal.
METHOD 2: Use of Normal Probability Plots
 A Normal Probability Plot is a scatter plot mapping the observation x to its z score.
 Some software has x on the x-axis, while some put z on the x-axis. Either way
you are looking for the same thing.
 If the points lie close to a straight line, the plot indicates that the data are
Normal. Systematic deviations from a straight line indicate a non-Normal
distribution. Outliers appear as points that are far away from the overall
pattern.
NORMAL PROBABILITY PLOTS ON THE GRAPHING CALCULATOR SEE PAGE 153 IN TEXTBOOK
Exercise 2.40 (page 156)
Exercise 2.41 (page 156)
SIMULATION:
HW: pg. 147; 2.36-2.39, pg.159; 2.50