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Chapter Six
Normal Curves and
Sampling Probability
Distributions
Chapter 6
Section 1
Graphs of Normal
Probability Distributions
Properties of The
Normal Distribution

The curve is bell-shaped with the highest
point over the mean,  .
Properties of The Normal
Distribution

The curve is symmetrical about a vertical
line through  .
Properties of The Normal
Distribution

The curve approaches the horizontal axis
but never touches or crosses it.
Properties of The Normal
Distribution
Inflection Point 
 1
 Inflection Point
   1
The transition points between cupping
upward and downward occur


above
 +  and  –  .

The Empirical Rule
 68.2% 

 1

  1
Approximately 68.2% of the data values
lie within
one
deviation of the


 standard
mean.
The Empirical Rule
 95.4% 
  2

  2
Approximately 95.4% of the data
 values lie withintwo standard

deviations of the mean.
The Empirical Rule
 99.7% 
  3

  3
Almost all (approximately 99.7%) of the
 data values will be withinthree standard

deviations of the mean.
The Empirical Rule
0.3410 0.3410
0.0015
0.0215
0.1360
  3   2   

0.1360
 
0.0215
0.0015
  2   3
Percentages of data that lies
between
given values

Application of the
Empirical Rule
Each of the variables in the left hand column of the table has a
normal probability distribution with the given mean ( ) and
standard deviation ( ). Use the empirical rule to complete the table.


68.2% fall
between
95.4% falls
between
99.7% fall
between
Height of adult
females
65”
2.5”
62.5  67.5
60  70
57.5  72.5
Contents of a box
of cereal
20 oz.
0.2 oz
19.8  20.2
19.6  20.4
19.4  20.6
Life span of a
battery
1000 hours
50 hours
950  1050
900 1100
850  1150
Diameter of an
engine part
3”
0.05”
2.95  3.05
2.9  3.1
2.85  3.15
Variable
Application of the
Empirical Rule
The life of a particular type
of light bulb is normally
distributed with a mean of
1100 hours and a standard
deviation of 100 hours.
Question: What is the
probability that a light
bulb of this type will
last between 1000 and
1200 hours?
P 1000  x  1200 
P 1100  100  x  1100  100 
P   1  x    1 
0.682
Answer:
Approximately
0.6820
Application of the
Empirical Rule
The life of a particular type
of light bulb is normally
distributed with a mean of
1100 hours and a standard
deviation of 75 hours.
Question: What is the
probability that a light
bulb of this type will
last between 950 and
1325 hours?
P 950  x  1325 
P 1100  150  x  1100  225 
P 1100  2 75   x  1100  375 
P   2  x    3 
0.1360  .3410  .3410  .1360  .0215
0.9755
Answer:
Approximately
0.9755
Application of the
Empirical Rule
I.Q. is normally distributed with   100 and  =15. Fill in the values that
correpsond 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
P 100  x  115 
P 100  x  100  15 
P 100  x  100  115 
P   x    1 
0.3410
Answer:
Approximately
0.3410
Application of the
Empirical Rule
I.Q. is normally distributed with   100 and  =15. Fill in the values that
correpsond 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.
b. Between 85 and 130
P 85  x  130 
P 100  15  x  100  30 
Answer:
P 100  115   x  100  2 15 
Approximately
P   1  x    2 
0.8180
0.3410  0.3410  0.1360
0.8180
Application of the
Empirical Rule
I.Q. is normally distributed with   100 and  =15. Fill in the values that
correpsond 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.
c. Between 130 and 145
P 130  x  145 
P 100  30  x  100  45 
Answer:
P 100  2 15   x  100  315 
Approximately
P   2  x    3 
0.0215
0.0215
Application of the
Empirical Rule
I.Q. is normally distributed with   100 and  =15. Fill in the values that
correpsond 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.
P x  130 
d. Over 130
P x  100  30 
P x  100  2 15 
P x    2 
0.0215  0.0015
0.0230
Answer:
Approximately
0.0230
Application of the
Empirical Rule
I.Q. is normally distributed with   100 and  =15. Fill in the values that
correpsond 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.
e. Under 55
P x  55 
P x  100  45 
P x  100  315 
P x    3 
0.0015
Answer:
Approximately
0.0015
Control Chart
A statistical tool to track data over
a period of equally spaced time
intervals or in some sequential
order.
Statistical Control
A random variable is in statistical control
if it can be described by the same
probability distribution when it is
observed at successive points in time.
To Construct a
Control Chart
• Draw a center horizontal line at  .
• Draw dashed lines (control limits) at
  2  and   .
• The values of   and  may be target values
or may be computed from past data when the
process was in control.
• Plot the variable being measured using time
on the horizontal axis.
Control Chart
 
 2

 2
 
Out-Of-Control
Warning Signals
I.
One point beyond the 3σ level.
II.
A run of nine consecutive points on one
side of the center line.
III. At least two of three consecutive points
beyond the 2σ level on the same side of
the center line.
Is the Process in Control?
 
 2

 2
 
Is the Process in Control?
 
 2

 2
 
Is the Process in Control?
 
 2

 2
 
Is the Process in Control?
 
 2

 2
 
Is the Process in Control?
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
Diameter
2.9
(inches)
2
3
4
5
6
7
8
9
10
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?
Is the Process in Control?
 
 2

 2
 
Is the Process in Control?
a. Do any points fall beyond the LCL and UCL three standard
deviation limits?
No points fall beyond the LCL and the UCL three standard
deviations limit.
•
Is there a run of nine consecutive points on one side of the center
line?
There is no run of nine consecutive points on one side of the
center line.
•
Is there an instance of two out of three points beyond the two
standard deviation limits on the same side of the center line?
There is no instance of two out of three points beyond the two
standard deviation limits on the same side of the center line.
Uniform Probability
Distributions
1
.
 
 
2. The mean is:  =
.
2
1. The equation is: y 
3. The standard deviation is:  =
 
12
.
ba
4. P a  x  b  
.
 
y
1
 

a
b

Uniform Probability
Distributions
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?
Uniform Probability
Distributions
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?
0.97  0.44

2
1.41

2
  0.705
Uniform Probability
Distributions
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%.
b. What is the standard deviation of the grades?
0.97  0.44

12
0.53

3.4641
  0.1530
Uniform Probability
Distributions
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%.
c. What is the probability of getting a grade between 65% and 75%?
P 0.65  x  0.75 
0.75  0.65
0.97  0.44
0.10
0.53
0.1887
Uniform Probability
Distributions
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%.
d. What is the probability of getting a grade 80% or higher?
P 0.80  x  0.97 
0.97  0.80
0.97  0.44
0.17
0.53
0.3208
Exponential Probability
Distributions
1. The equation is: y 
1


x
e .
2. The mean is:  = .
3. The standard deviation is:  = .
4. P a  x  b   e

a


b
e .
1

y
a
1


e
b
x

Uniform Probability
Distributions
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?
Uniform Probability
Distributions
The intersection in downtown Annville is experiencing an accident about
every 40 days.
a. What is the mean number of days between accidents?
  40
Uniform Probability
Distributions
The intersection in downtown Annville is experiencing an accident about
every 40 days.
b. What is the standard deviation of the number of days between
accidents?
  40
Uniform Probability
Distributions
The intersection in downtown Annville is experiencing an accident about
every 40 days.
c. What is the probability of having another
accident after 30 to 60 days?
P 30  x  60 
e

30
40
0.75
e

60
40
1.5
e
e
0.4724  0.2231
0.2492
Uniform Probability
Distributions
The intersection in downtown Annville is experiencing an accident about
every 40 days.
P x  60 
d. What is the probability of having another
accident after more than 60 days?
e
60

40
1.5
e


40

e e
0.2231 0
0.2231
THE END
OF
SECTION 1
Homework Assignments
Pages: 259 - 266
Exercises: #1 - 19, odd
Exercises: #2 - 20, even