Download ESTIMATE areas (proportions of values) in a Normal

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AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Objectives:
• ESTIMATE areas (proportions of values) in a Normal distribution.
• FIND the proportion of z-values in a specified interval, or a z-score from a
percentile in the standard Normal distribution.
• FIND the proportion of values in a specified interval, or the value that
corresponds to a given percentile in any Normal distribution.
• DETERMINE whether a distribution of data is approximately Normal from
graphical and numerical evidence.
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
All Normal distributions are the same if we measure in units of size
σ from the mean µ as center.
The standard Normal distribution is the Normal distribution
with mean 0 and standard deviation 1. N(0, 1)
When we find z-scores (standardize a value), we are transforming
our data so that it fits on a standard normal distribution. Remember
that this allows us to compare distributions of different sizes, or
different measures, etc.
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
TABLE A at the back of your book (pages T1 and T2 is called the
STANDARD NORMAL TABLE. This is a table of areas under
the standard normal curve. The entry in the table for each value of z
gives you the area under the curve to the LEFT of z. (Remember percentiles are also the area to the left of the given value.)
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AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Suppose we want to find the
proportion of observations
from the standard Normal
distribution that are less than
0.81.
We can use Table A:
Z
-
scores
are
the
on
outside edge Area
inside
Is on the
.
.
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
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AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
be
* each number should
listed and Labeled if you
use
normal
cdf
.
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Example: Suppose we have a Normal distribution, find the
following proportions using Table A, then check them on your
calculator.
a)
b)
Find the proportion of observations to the left of z=-1.41.
.org#q
II.
Table
:
Area
.org
=
Normal cdf ( mints
Max
,
=
-1.41 ,
no
,
at
)
=
Find the proportion of observations less than 3.56.
Table
:D
Normal cdf (
.998
Coftthetabkateotuhenatopggo
,
mm
=
-
a
,
Max =3 56
.
,µ=o
,
0=1
)
=
xD
.
99981
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Example: Suppose we have a Normal distribution, find the
following proportions using Table A, then check them on your
calculator.
c) The proportion of the distribution to the right of -2.13.
¥-4
d)
Area =L
area
-
Normal
to the
cdf (
min
Left
=
-
I
=
2.13
,
0166
-
.
Max
=
a
.ci#
=
,µ=l
,
0=0
)
=
[email protected]
The proportion of the distribution greater than .08.
'
Area
88k€
=
de
Normal
-
It
cdf (
=
OR
minMax
=
A
,µ=o
,
0=1
)
=
[email protected]
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Example: Suppose we have a Normal distribution, find the following
proportions using Table A, then check them on your calculator.
e) Find the proportion of observations between z=-.03 and
0<3*99-53)
f)
Area
-_
9953
.
-
.
3821=-61320
z=2.6.
Find the proportion of observations between z=.23 and
z=3.45.
t.nl#ENormalcdf(mm=-.3,max=2.6,m=o,o=l)[email protected][email protected]*MgysN0rma1cdf(mM=.23,max=3.4s,n=o,[email protected]
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Example: The scores of a reference population on the Wechsler Intelligence Scale
for Children (WISC) are normally distributed with r=100 and =15.
A school district classified children as “gifted” if their WISC score exceeds 135.
There are 1300 sixth graders in the school district. About how many of them are
gifted?
o
=X
Z
die
=
135,4
-
or
=
mm
=
135
#
of gifted Kids
0099113001=12.87
12
.
T
gifted
students
are
=
1-
.
9901
0099
=
.
-
Normal cdf (
=
Area
2.33
considered
,
=
Max
,µ=
a
100,0=15)
=
.
0098
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Working Backwards: Sometimes we're given a percentile and we
need to find the z-score.
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AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Examples: (double check your answers using the Normalcdf or invNorm
functions on your calculator)
a) Find the number z such that the proportion of observations that are less than
z in a standard normal distribution is 0.98.
mvNorm(area=98,µ=O,o=1 )
or
.tn?afo3tabkatz=#
-9M¥ Ynoektdrsast
2=2.0540
.
b) Find the number z such that 22% of all observations from a standard
distribution are greater than z.
<¥
IM
normal
[email protected][email protected]
orz=mvNorm(
Look
for
-78
on
the
area
-78
,M= 0,0=11
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Assessing the Normality of a Distribution:
The Normal distributions provide good models for some
distributions of real data.
Many statistical inference procedures are based on the assumption
that the population is approximately Normally distributed.
A Normal probability plot provides a good assessment of
whether a data set follows a Normal distribution.
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
METHOD 1: Use all of the methods from Chapter 1 in
combination.
1. Construct a stemplot or histogram. Is it bell shaped and
symmetric around the mean?
2.
Find the mean and standard deviation.
3. Draw a normal distribution graph and mark the values at the
mean, then at 1 and 2 standard deviations away.
4.
Check the 68-95-99.7 rule. You can do this two ways standardize the data and find areas, or compare the relative
frequency of the data in each section.
NOTE: the smaller the set of data, the less likely it is to follow the
rule exactly. You may have to make a judgment call then proceed
with caution.
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
METHOD 2:
Construct a NORMAL PROBABILITY PLOT
on your calculator. If the graph looks linear, it means that our
distribution is normal. If it’s a concave curve, the distribution is
skewed left. If it’s a convex curve, the distribution is skewed right.
Outliers will appear as points that are far away from the overall
pattern.
Step 1: Enter your data in list 1
Step 2: Define your statplot as below, then use Zoom 9:Stat to see
the plot
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Example: The measurements listed below describe the usable
capacity (in cubic feet) of a sample of 36 side-by-side refrigerators.
Are the data close to normal? Show using both methods.
12.9
15.2
16.0
16.6
13.7
15.3
16.0
16.8
14.1
15.3
16.2
17.0
14.2
15.3
16.2
17.0
14.5
15.3
16.3
17.2
14.5
15.5
16.4
17.4
14.6
15.6
16.5
17.4
14.7
15.6
16.6
17.9
15.1
15.8
16.6
18.4
method
[email protected]
'
.
#
*
z
÷÷mdIy¥M¥I¥Eiiµwn±¥MY"
I:*
I=
15.82
S×=
1.21
within ±2sy
:
34/36=94.4 't
:
.
:
.
.
Within'=3s×
36/36=1001
:
I
iii.
.
Pretty linear
.
Approximately Normal
.
AP Stats ~ 2D: Standard Normal Distributions, and assessing normality
Homework:
page 129: #47-53, 55-67 odds,
72-74