Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 The Normal Distribution

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1

The Normal Distribution


Properties of The Normal Distribution

The curve is bell-shaped with the highest point over the mean, .



The curve is symmetrical about a vertical line through .



The curve approaches the horizontal axis but never touches or crosses it.



The transition points between cupping upward and downward occur

above + and – .

–


The Empirical Rule

Approximately 68% of the data values lie is within one standard deviation of the mean.

One standard deviation from the mean.

68%


The Empirical RuleThe Empirical Rule

Approximately 95% of the data values lie within two standard deviations of the mean.

Two standard deviations from the mean.

x95%


The Empirical RuleThe Empirical Rule

Almost all (approximately 99.7%) of the data values will be within three standard deviations of

the mean.

Three standard deviations from the mean.

x99.7%


Section 6.1, Exercise 7: Assuming that the heights of collegewomen are normally distributed with mean 65in. and standarddeviation 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.


Example (similar to Section 6.1, Exercise 10). Suppose A vending machine automatically pours decaf coffee into10oz 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 the10oz 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 notoverflow when served?



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

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

1 2 3 4 5 6 7


Control Chart

1 2 3 4 5 6 7


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.


Probability of a False Alarm

Warning Signal Probability of falsealarm

I Point beyond 3 0.00

IINineconscecutivepointsonsamesideof

0.004

IIIAtleast/pointsbeyond

0.004


Is the Process in Control?

1 2 3 4 5 6 7



1 2 3 4 5 6 7 8 9 10 11 12 13



1 2 3 4 5 6 7



1 2 3 4 5 6 7

Documents

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 The Normal Distribution