Download Probability and Statistics

Probability and Statistics
Normal Distributions
Chapter 6
Section 3
Area Under Any Normal Curve
Essential Question: What is the process for determining the area under a normal curve?
Student Objectives: The students will calculate the probability of a “standardized” event.
The students will find a z-score given the normal probability (inverse
normal).
The student will use the inverse normal to solve a guarantee problem.
Terms:
Inverse normal
Pearson’s Index
Standard Normal Curve
z-score
Key Concepts:
Converting any normally distribution into a standard normal distribution. This allows the
use of one table of values to determine the all the probabilities.
Check for Normality
1. Draw a histogram. For a normal distribution, the histogram should be roughly
bell-shaped.
2. Outliers. for a normal distribution, there should be no more than one outlier
on a box-n-whisker plot.
3. Skewness. Normal distributions are symmetric. Use Pearson’s index for
skewness. A score smaller than negative 1 or larger than 1 indicates the data
is skewed.
4. Normal quartile plot. This graph plots points as (z-score, x-value). The data
is considered NOT to be skewed if the points lie in a straight line.
Equations:
z-score: z =
x !µ
"
Pearson's Index of Skewness
!1 "
3( x ! median )
"1
s
Graphing Calculator Skills:
2nd VARS DISTR Normalcdf(lower bound, upper bound, µ , ! )
2nd VARS DRAW ShadeNorm(lower bound, upper bound, µ , ! )
2nd VARS DISTR invNorm(area, µ , ! )
2nd STAT PLOT; ON, Type (6th option; Data: L; Data Axis Y; Mark: “box”
Sample Questions:
1. If x is a normally distributed variable with a mean of 30 and a standard deviation of 6,
find the following probabilities:
a.
P ( x ! 30 )
b.
P ( x ! 18 )
c.
P ( x ! 36 )
d.
P ( 24 ! x ! 39 )
2. Determine the z scores that produce the following probabilities.
a. 0.20 lies to the right of the z-score
b. 0.40 lies to the right of the z-score
c. 0.875 lies to the right of the z-score.
d. 0.6328 lies between z and -z.
3. Consider the following data set: {1, 5, 2, 3, 4, 3, 3, 4, 4, 3, 2} to answer the following
questions.
a.
Make a histogram for this data
b.
Make a Box-n-Whisker plot for the data.
<----+----+----+----+----+----+----+----+----+---->
c.
What is the Pearson’s Index for the set of data?
d.
Make a normal quartile plot for the data
e.
Interpret the results from parts a through d.
Homework: Pages 286 - 291
Exercises: 1, 5, 9, 13, 17, 21, 25, 29, 33, and 37
Exercises: 3, 7, 11, 15, 19, 23, 27, 31, 35, and 39
Exercises: 2, 6, 10, 14, 18, 22, 26, 30, 34, and 38
Exercises: 4, 8, 12, 16, 20, 24, 28, 32, 36, and 40