Download TUTORIAL 8 - Normal Distributions

Statistics Workshop
Tutorial 8
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Normal Distributions
Slide 2
Section 5-1
Overview
Created by Erin Hodgess, Houston, Texas
Copyright © 2004 Pearson Education, Inc.
LAPTOP3:
Overview
Slide 3
 Continuous random variable
 Normal distribution
-1
x-
(
)
2
f(x) =
e

2
2p
Formula 5-1
Figure 5-1
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Slide 4
Section 5-2
The Standard Normal
Distribution
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Definitions
 Uniform Distribution
Slide 5
is a probability
distribution in which the continuous
random variable values are spread
evenly over the range of possibilities;
the graph results in a rectangular shape.
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Definitions
Slide 6
 Density Curve (or probability density
function is the graph of a continuous
probability distribution.
1. The total area under the curve must
equal 1.
2. Every point on the curve must have a
vertical height that is 0 or greater.
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Slide 7
Because the total area under
the density curve is equal to 1,
there is a correspondence
between area and probability.
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Using Area to
Find Probability
Figure 5-3
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Slide 8
Heights of Adult Men and Women
Slide 9
Figure 5-4
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Definition
Slide 10
Standard Normal Distribution:
a normal probability distribution that has a
mean of 0 and a standard deviation of 1.
Figure 5-5
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Slide 11
Example:
If thermometers have an average (mean)
reading of 0 degrees and a standard deviation of 1 degree
for freezing water and if one thermometer is randomly
selected, find the probability that, at the freezing point of
water, the reading is less than 1.58 degrees.
P (z < 1.58) = 0.9429
The probability that the chosen thermometer will measure
freezing water less than 1.58 degrees is 0.9429.
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Slide 12
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Slide 13
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Finding z Scores
when Given Probabilities
Slide 14
5% or 0.05
(z score will be positive)
1.645
Figure 5-10
Finding the 95th Percentile
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Slide 15
Section 5-3
Applications of Normal
Distributions
Created by Erin Hodgess, Houston, Texas
Copyright © 2004 Pearson Education, Inc.
Nonstandard Normal
Distributions


Slide 16
If
 0 or  1 (or both), we will
convert values to standard scores using
Formula 5-2, then procedures for working
with all normal distributions are the same
as those for the standard normal
distribution.
Formula 5-2
z=
x–µ

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Converting to Standard
Normal Distribution
z=
x–

Figure 5-12
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Slide 17
Probability of Weight between Slide 18
140 pounds and 211 pounds
In the Chapter Problem, we noted that the Air Force
had been using the ACES-II ejection seats designed
for men weighing between 140 lb and 211 lb. Given
that women’s weights are normally distributed with a
mean of 143 lb and a standard deviation of 29 lb
(based on data from the National Health survey),
what percentage of women have weights that are
within those limits?
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Probability of Weight between Slide 19
140 pounds and 211 pounds
 = 143
 = 29
There is a 0.5302 probability of randomly
selecting a woman with
a weight between 140 and 211 lbs.
(53.02% of women)
Figure 5-14
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Find P98 for Hip
Breadths of Men
x =  + (z ● )
x = 14.4 + (2.05  1.0)
x = 16.45
Slide 20
The hip breadth of 16.5 in. separates
the lowest 98% from the highest 2%
Seats designed for a
hip breadth up to 16.5
in. will fit 98% of men.
Figure 5-15
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Slide 21
Section 5-4
Sampling Distributions
and Estimators
Created by Erin Hodgess, Houston, Texas
Copyright © 2004 Pearson Education, Inc.
Definition
Slide 22
Sampling Distribution of the mean
is the probability distribution of
sample means, with all
samples having the same sample
size n.
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Definitio
n
Sampling Variability:
Slide 23
The value of a statistic, such as the
sample mean x, depends on the
particular values included in the
sample.
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Slide 24
Section 5-5
The Central Limit
Theorem
Created by Erin Hodgess, Houston, Texas
Copyright © 2004 Pearson Education, Inc.
Central Limit Theorem
Slide 25
Given:
1. The random variable x has a distribution (which
may or may not be normal) with mean µ and
standard deviation .
2. Samples all of the same size n are randomly
selected from the population of x values.
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Central Limit Theorem
Slide 26
Conclusions:
1. The distribution of sample x will, as the
sample size increases, approach a normal
distribution.
2. The mean of the sample means will be the
population mean µ.
3. The standard deviation of the sample means
n
will approach 
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Practical Rules
Commonly Used:
Slide 27
1. For samples of size n larger than 30, the distribution of
the sample means can be approximated reasonably well
by a normal distribution. The approximation gets better
as the sample size n becomes larger.
2. If the original population is itself normally distributed,
then the sample means will be normally distributed for
any sample size n (not just the values of n larger than 30).
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Notation
Slide 28
the mean of the sample means
µx = µ
the standard deviation of sample mean

x = n
(often called standard error of the mean)
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Slide 29
Section 5-6
Normal as Approximation
to Binomial
Created by Erin Hodgess, Houston, Texas
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Approximate a Binomial Distribution
Slide 30
with a Normal Distribution if:
np  5
nq  5
then µ = np and  =
npq
and the random variable has
a
distribution.
(normal)
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Finding the Probability of
Slide 31
“At Least”
120 Men Among 200 Accepted Applicants
Figure 5-24
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Definition
Slide 32
When we use the normal distribution
(which is continuous) as an
approximation to the binomial
distribution (which is discrete), a
continuity correction is made to a
discrete whole number x in the binomial
distribution by representing the single
value x by the interval from
x – 0.5 to x + 0.5.
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x = at least 120
= 120, 121, 122, . . .
x = more than 120
= 121, 122, 123, . . .
x = at most 120
= 0, 1, . . . 118, 119, 120
x = fewer than 120
= 0, 1, . . . 118, 119
Figure 5-25
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Slide 33
x = exactly 120
Interval represents discrete number 120
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Slide 34
Slide 35
Section 5-7
Determining Normality
Created by Erin Hodgess, Houston, Texas
Copyright © 2004 Pearson Education, Inc.
Definition
Slide 36
A Normal Quantile Plot is a graph of
points (x,y), where each x value is
from the original set of sample
data, and each y value is a z score
corresponding to a quantile value
of the standard normal distribution.
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Procedure for Determining
Whether Data Have a
Normal Distribution
Slide 37
1. Histogram: Construct a histogram. Reject
normality if the histogram departs dramatically
from a bell shape.
2. Outliers: Identify outliers. Reject normality if there
is more than one outlier present.
3. Normal Quantile Plot: If the histogram is basically
symmetric and there is at most one outlier,
construct a normal quantile plot as follows:
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Procedure for Determining
Whether Data Have a
Normal Distribution
Slide 38
Examine the normal quantile plot and
reject normality if the the points do not
lie close to a straight line, or if the points
exhibit some systematic pattern that is
not a straight-line pattern.
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Now we are ready for
Day 3