Transcript Chapter 7

Normal
Distributions
1
Normal Distributions
1.
2.
3.
4.
Symmetrical bell-shaped (unimodal) density curve
How is this done
Above the horizontal axis
mathematically?
N(m, s)
The transition points occur at m + s (Points of
inflection)
5. Probability is calculated by finding the area
under the curve
6. As s increases, the curve flattens &
spreads out
7. As s decreases, the curve gets
2
taller and thinner
Normal distributions occur
frequently.
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•
Length of newborns
Height
Weight
ACT or SAT scores
Intelligence
Number of typing errors
Chemical processes
3
A
6
B
s
s
Do these two normal curves have the same mean?
If so, what is it?
YES
Which normal curve has a standard deviation of 3?
B
Which normal curve has a standard deviation of 1?
A
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Empirical Rule
• Approximately 68% of the
observations fall within s of m
• Approximately 95% of the
observations fall within 2s of m
• Approximately 99.7% of the
observations fall within 3s of m
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Suppose that the height of male
students at JVHS is normally
distributed with a mean of 71 inches
and standard deviation of 2.5 inches.
What is the probability that the
height of a randomly selected male
student is more than 73.5 inches?
1 - 0.68 = 0.32
P(X > 73.5) = 0.16
68%
71
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Standard Normal Density
Curves
Always has m = 0 & s = 1
To standardize:
z 
x m
s
Must have
this
memorized!
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Strategies for finding probabilities
or proportions in normal
distributions
1. State the probability
statement
2. Draw a picture
3. Calculate the z-score
4. Look up the probability
(proportion) in the table
8
AP Statistics
Friday, 09 January 2015
•
OBJECTIVE TSW investigate normal distributions.
•
You need to have the following out:
1.
2.
Blue chart (Table A)
Calculator (sign up for a new/old number, if needed)
• QUIZ: Continuous & Uniform Distributions on
Monday, 12 January 2015.
• ASSIGNMENT DUE DATES
–
–
–
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WS Normal Distributions  Monday, 01/12/2015
WS Continuous Distributions Review  Tuesday, 01/13/2015
Bookwork: 7.67, 7.69, 7.71  Monday, 01/12/2015
Bookwork: 7.73, 7.77, 7.80  Monday, 01/12/2015
WS Unusual Density Curves
1)
2)
3)
4)
5)
6)
7)
8)
9)
0.375
0.375
0.34375
0.5
0.35
0.3
0.55
0.8125
a)
A = 0.05(0.4)(5) = 1
b)
P(X < 0.20) = 50%
P(X < 0.10) = 12.5%
c)
37.5%
d)
43.75%
WS Uniform Distributions
1)
a)
b)
μ = 2 min, σ = 1.15470 min
0.375
2)
a)
b)
10 min
0.05
3)
a)
b)
c)
0.666
0.333
82.5 degrees
4)
a)
b)
c)
d)
e)
continuous
μ = 7 oz., σ = 0.28868 oz.
0.75
0.25
0.57735
5)
0.00024414
The lifetime of a certain type of battery
is normally distributed with a mean of
200 hours
and
a standardDraw
deviation
of 15
& shade
Write
the
hours. What
proportion of these
the curve
probability
batteries
can be expected to last less
statement
than 220 hours?
P(X < 220) = 0.9082
Look up z220
 200
score
in
z 
 1.33
table
15
Calculate z-score
12
The lifetime of a certain type of battery
is normally distributed with a mean of
200 hours and a standard deviation of 15
hours. What proportion of these
batteries can be expected to last more
than 220 hours?
P(X>220) = 1 - 0.9082
= 0.0918
220 200
z 
 1.33
15
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The lifetime of a certain type of battery
is normally distributed with a mean of
200 hours and a standard deviation of 15
Look
up in
table 0.95
hours. How long
must
a battery
last to be
in the top 5%? to find z- score
P(X > ?) = 0.05
x  200
1.645 
15
x  224.675
224.675 hours
.95
.05
1.645
 Label units when given
14
The heights of the female students at
JVHS are normally distributed with a
What
is the zmean of 65 inches. What
is the
for the
standard deviation of this score
distribution
63?
if 18.5% of the female students are
shorter than 63 inches?
P(X < 63) = 0.185
-0.9
63  65
0.9 
s
63
2
s
 2.2222 inches
0.9
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The heights of female teachers at JVHS
are normally distributed with mean of 65.5
inches and standard deviation of 2.25
inches. The heights of male teachers are
normally distributed with mean of 70 inches
and standard deviation of 2.5 inches.
• Describe the distribution of differences of
heights (male – female) teachers.
m = 70 - 65.5
m = 4.5
s = 2.252 + 2.52
s = 3.36340...
Normal distribution with
m = 4.5 inches &
s = 3.3634 inches
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• What is the probability that a
randomly selected male teacher is
shorter than a randomly selected
female teacher?
P(X<0) = 0.0901
0  4.5
z 
 1.34
3.3634
4.5
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Will my calculator do any
of this normal stuff?
• Normalpdf – use for graphing ONLY
• Normalcdf – will find probability of
area from lower bound to upper
bound
• Invnorm (inverse normal) – will find
X-value for probability
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Assignments
• WS Normal Distributions
– Due on Monday, 12 January 2015.
• WS Continuous Distributions Review
– Due on Tuesday, 13 January 2015.
• Bookwork: 7.67, 7.69, 7.71, 7.73, 7.77, 7.80
– Due on Monday, 12 January 2015.
• QUIZ: Continuous & Uniform Distributions on
Monday, 12 January 2015.
19
AP Statistics
Monday, 12 January 2015
•
OBJECTIVE TSW quiz over continuous and uniform
•
READ
•
•
distributions.
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Sec. 7.7: pp. 409-416
ASSIGNMENTS DUE
WS Normal Distributions  wire basket
Bookwork: 7.67, 7.69, 7.71, 7.73, 7.77, 7.80
ASSIGNMENT DUE TOMORROW
WS Continuous Distributions Review
 black tray
Ways to Assess Normality
1. Use graphs (dotplots,
boxplots, or histograms)
2. Use the Empirical Rule
3. Normal probability
(quantile) plot
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Normal Probability (Quantile) plots
1. The observation (x) is plotted against
known normal z-scores
2. If the points on the quantile plot lie close
to a straight line, then the data is
normally distributed
3. Deviations on the quantile plot indicate
nonnormal data
4. Points far away from the plot indicate
outliers
5. Vertical stacks of points (repeated
observations of the same number) is called
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granularity
Normal Scores
Suppose
we have
the following
To
construct
a normal
probability plot,
Sketch
a
scatterplot
by
pairing
the
Think
of
selecting
sample
after
sample
of
observations
of
widths
of
contact
you cansmallest
use quantities
called
normal
normal
score with
the
size
10
from
a
standard
normal
windows
in integrated
circuit
chips:
What should
score.
The
values
of
the
normal
scores
smallest
observation
from
the
the
1distribution. Then -1.539 is data
happen
if sample size n. The normal
depend
on
the
set smallest
& so on observation
average of the
our when
data n = 10 are below:
scores
from each sample & so on . . .
is
3.21 set2.49
2.94 4.38 4.02
2
33.34
4 3.81
5
3.62normally
3.301 2.85
distributed?
-1
-1.539 -1.001 -0.656 -0.376 -0.123
Contact 1.001
Windows1.539
0.123 Widths
0.376of 0.656
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Are these approximately normally
distributed?
50 48 54 47 51 52 46 53
What
52 51 48 48 54 55
57is this
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53 50 47 49 50 56 called?
53 52
Both the histogram & boxplot
are approximately
symmetrical, so these data
are approximately normal.
The normal probability
plot is approximately
linear, so these data are
approximately normal. 24
QUIZ: Continuous and Uniform Distributions
• Checkerboard, please.
• ASSIGNMENT DUE TOMORROW
– WS Continuous Distributions Review
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