Download Introduction to the normal distribution

The Normal Distribution
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1
Properties of The Normal
Distribution

The curve is bell-shaped with the
highest point over the mean, .
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2
Properties of The Normal
Distribution

The curve is symmetrical about a
vertical line through .
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3
Properties of The Normal
Distribution

The curve approaches the horizontal
axis but never touches or crosses it.
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4
Properties of The Normal
Distribution
–


The transition points between cupping
upward and downward occur
above  +  and  –  .
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5
The Empirical Rule
Approximately 68% of the data values lie is
within one standard deviation of the mean.
68%

One standard deviation from the mean.
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6
The Empirical Rule
Approximately 95% of the data values lie within
two standard deviations of the mean.
95%
x
Two standard deviations from the mean.
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7
The Empirical Rule
Almost all (approximately 99.7%) of the data
values will be within three standard deviations of
the mean.
99.7%
x
Three standard deviations from the mean.
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8
Section 6.1, Exercise 7: Assuming that the heights of college
women are normally distributed with mean 65in. and standard
deviation 2.5in., answer the following questions.
(a) What percentage of women are taller than 65 in.?
(b) What percentage of women are shorter than 65in.?
(c) What percentage of women are between 62.5 in and 67.5in?
(d) What percentage of women are between 60in. and 70in?
Hint: use the empirical rule.
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9
Example (similar to Section 6.1, Exercise 10). Suppose
A vending machine automatically pours decaf coffee into
10oz capacity cups. It was found that the amount of decaf
dispensed is normally distributed with a mean of 9.5 oz.
and standard deviation of .25 oz.
(a) Estimate the probability the machine will overflow the
10oz cup.
(b) If the machine has been loaded with 1000 cups, how
many do you expect will overflow when served?
(c) If one thousand 9oz cups were mistakenly put in the
dispenser, how many do you expect would not
overflow when served?
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10
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11
Control Chart
a statistical tool to track data over a
period of equally spaced time
intervals or in some sequential
order
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12
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.
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13
To Construct a Control Chart
• Draw a center horizontal line at .
• Draw dashed lines (control limits) at
    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.
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14
Control Chart





1
2
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Control Chart





1
2
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Out-Of-Control Warning
Signals
I
One point beyond the 3 level
II
A run of nine consecutive points on
one side of the center line at target 
III
At least two of three consecutive
points beyond the 2 level on the same
side of the center line.
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17
Probability of a False Alarm
Warning Signal
Probability of false
alarm
I Point beyond 3
0.003
II Nine conscecutive
points on same side of

III At least 2/3 points
beyond 2
0.004
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0.004
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Is the Process in Control?





1
2
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Is the Process in Control?





1 2 3 4 5 6 7 8 9 10 11 12 13
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Is the Process in Control?





1
2
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Is the Process in Control?





1
2
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