Probability and Statistics
Normal Curves and Sampling Distributions
Chapter 6
Section 1
Graphs of Normal Probability Distributions
Essential Question: How is the normal curve used to determine the probability of an
event?
Student Objectives: The student will graph a normal curve and summarize its important
features.
The student will apply the empirical rule to solve “real-world”
problems.
The student will use the control limits to construct control charts.
The student will use the control charts to determine if the situation is
“out-of-control”.
The students will determine the probability of an event in a Uniform
Probability Distribution.
The student will determine the probability of an event in an
Exponential Distribution.
Terms:
Continuous random variable
Control Charts
Empirical Rule
Exponential Distribution
Normal curve - “bell-shaped curve”
Normal distribution (Gaussian)
Uniform Distribution
Key Points:
Properties of a Normal Curve:
1. The curve is bell-shaped with the highest point over the mean ( µ ).
2. It is symmetrical about the vertical line through the mean ( µ ).
3. The curve approaches the horizontal axis but never touches or crosses it.
4. The transition points between being concave downward and concave upward
occur at µ ± 1! .
The Empirical Rule:
For a distribution that is symmetrical and bell-shaped the following are true:
1. 68.2% of the data lies between µ ± 1! .
2. 95.4% of the data lies between µ ± 2! .
3. 99.7% of the data lies between µ ± 3! .
Control Charts - “out-of-Control Warning Signals”:
1. Out-of-Control Signal I: One point beyond the µ ± 3! level.
2. Out-of-Control Signal II: A run of nine consecutive points on one side of the
center line.
3. Out-of-Control Signal III: Two out three points are beyond the µ ± 2! level.
Uniform Probability Distribution:
1
1. The equation is: y =
.
! "#
! +"
2. The mean is: µ =
.
2
" #$
.
12
b"a
4. The probability: P ( a ! x ! b ) =
# "$
3. The standard deviation is: ! =
Exponential Probability Distribution:
x
1 "!
1. The equation is: y = e .
!
2. The mean is: µ = ! .
3. The standard deviation is: ! = " .
4. The probability: P ( a ! x ! b ) = e
"
a
#
"e
"
b
#
.
5. The non-bounded probability: P ( x ! a ) = e
"
a
#
.
Graphing Calculator Skills:
Drawing and shading normal curves
Drawing Control Charts
Drawing the graph of a Uniform Probability Distribution
Calculating the probability of an event in a Uniform Probability Distribution
Drawing an Exponential Probability Distribution.
Calculating the probability of an event in an Exponential Probability Distribution.
Sample Questions:
1.
You are in charge of Quality Control for a manufacturing company that produces
parts for automobiles. A specific gear has been designed to have a diameter of three
inches. We have learned from that the standard deviation of the gear is 0.2 inches.
The following ten measurements were taken from a random sample of gears that came
off the production line. Make a control chart on graph paper for the measures given
below. Does this indicate that the measures are in control?
Part
1
2
3
4
5
6
7
8
9
10
Diameter
(inches)
2.9
2.6
3.1
3.5
2.8
2.9
3.4
3.2
2.7
3.3
a.
Do any points fall beyond the LCL and UCL three standard deviation limits?
b.
Is there a run of nine consecutive points on one side of the center line?
c.
Is there an instance of two out of three points beyond the two standard
deviation limits on the same side of the center line?
2.
Each of the variables in the left hand column of the table has a normal probability
distribution with the given mean ( µ ) and standard deviation ( ! ). Use the empirical
rule to complete the table.
Variable
µ
!
Height of
adult
females
65”
2.5”
Contents of
a box of
cereal
20 oz.
0.2 oz
Life span
1000 hours
of a battery
Diameter
of an
engine part
3”
50 hours
0.05”
68.2% fall
between
95.4% falls
between
99.7% fall
between
3.
I.Q. is normally distributed with µ = 100 and ! = 15 . Fill in the values that
correspond to the standard deviation marks on the number line and find the
probability that a person picked at random out of the general population has an I.Q. in
the general interval.
a.
Between 100 and 115
b.
Between 85 and 130
c.
Between 130 and 145
d.
Over 130
e.
Less than 55
4.
A professor noticed that the grades for his final examination fit a Uniform Probability
Distribution where the highest grade was a 97% and the lowest grade was a 44%.
a.
What is the mean grade?
b.
What is the standard deviation of the grades?
c.
What is the probability of getting a grade between 65% and 75%?
d.
What is the probability of getting a grade 80% or higher?
5.
The intersection in downtown Annville is experiencing an accident about every 40
days.
a.
What is the mean number of days between accidents?
b.
What is the standard deviation of the number of days between accidents?
c.
What is the probability of having another accident after 30 to 60 days?
d.
What is the probability of having another accident after more than 60 days?
Homework Assignments:
P ages: 259 - 266
Exercises: #1 - 19, odd
Exercises: #2 - 20, even