Download Normal Distribution

STAT 101
Dr. Kari Lock Morgan
10/18/12
Normal Distribution
Chapter 5
• Normal distribution
• Central limit theorem
• Normal distribution for confidence intervals
• Normal distribution for p-values
• Standard normal
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Project 1
• Due Tuesday
• 5 pages, double spaced, including figures
• Hypotheses should not change based on data
• This is a research paper – there should be
text and complete sentences.
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Bootstrap and Randomization Distributions
Correlation: Malevolent
uniforms
Measures from Scrambled Collection 1
Slope :Restaurant
tips
Measures from Scrambled RestaurantTips
-60
-40
Dot Plot
-20
0
20
slope (thousandths)
Mean :Body
Temperatures
Measures from Sample of BodyTemp50
98.2
98.3
98.4
40
-0.4
-0.2
0.0
r
0.2
All
bell-shaped
What
do you
Diff means: Finger taps
distributions!
notice?
0.4
0.6
Dot Plot
Measures from Scrambled CaffeineTaps
98.5
98.6
Nullxbar
98.7
98.8
0.5
phat
0.6
98.9
Dot Plot
Dot Plot
99.0
-4
Proportion : Owners/dogs
0.4
60
-0.6
Measures from Sample of Collection 1
0.3
Dot Plot
-3
-2
-1
0
Diff
1
2
3
Mean : Atlanta commutes
Measures from Sample of CommuteAtlanta
0.7
0.8
Statistics: Unlocking the Power of Data
26
27
28
29
xbar
30
4
Dot Plot
31
32
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Normal Distribution
• The symmetric, bell-shaped curve we have
1000
0
500
Frequency
1500
seen for almost all of our bootstrap and
randomization distributions is called a
normal distribution
-3
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-2
-1
0
1
2
3
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Central Limit Theorem!
For a sufficiently large sample
size, the distribution of sample
statistics for a mean or a
proportion is normal
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Central Limit Theorem
• The central limit theorem holds for ANY
original distribution, although “sufficiently large
sample size” varies
• The more skewed the original distribution is
(the farther from normal), the larger the sample
size has to be for the CLT to work
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Central Limit Theorem
• For distributions of a quantitative variable that
are not very skewed and without large outliers,
n ≥ 30 is usually sufficient to use the CLT
• For distributions of a categorical variable,
counts of at least 10 within each category is
usually sufficient to use the CLT
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Normal Distribution
• The normal distribution is fully
characterized by it’s mean and standard
deviation
N  mean,standard deviation 
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Normal Distribution
N  0.523,0.048
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Bootstrap Distributions
If a bootstrap distribution is
approximately normally distributed, we
can write it as
a)
b)
c)
d)
N(parameter, sd)
N(statistic, sd)
N(parameter, se)
N(statistic, se)
sd = standard deviation of variable
se = standard error = standard deviation of statistic
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Confidence Intervals
If the bootstrap distribution is normal:
To find a P% confidence interval , we just
need to find the middle P% of the
distribution
N(statistic, SE)
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Best Picture
What proportion of
visitors to
www.naplesnews.com
thought The Artist
should win best
picture?
pˆ  .15
SE  ???
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Best Picture
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Area under a Curve
• The area under the curve of a normal
distribution is equal to the proportion of the
distribution falling within that range
• Knowing just the mean and standard
deviation of a normal distribution allows
you to calculate areas in the tails and
percentiles
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Best Picture
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Best Picture
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Confidence Intervals
For a normal sampling distribution, we
can also use the formula
sample statistic  2  SE
to give a 95% confidence interval.
.156  2  .03
 0.096, 0.216 
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Confidence Intervals
For normal bootstrap distributions, the
formula
sample statistic  2  SE
gives a 95% confidence interval.
How would you use the N(0,1) normal
distribution to find the appropriate
multiplier for other levels of confidence?
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Confidence Intervals
For a P% confidence interval, use
sample statistic  z  SE
*
where P% of a N(0,1) distribution is
between –z* and z*
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Confidence Intervals
P%
-z*
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z*
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Confidence Intervals
Find z* for a 99% confidence interval.
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z* = 2.575
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News Sources
“A new national survey shows that the
majority (64%) of American adults use at
least three different types of media every
week to get news and information about
their local community”
The standard error for this statistic is 1%
Find a 99% confidence interval for the
true proportion.
Source: http://pewresearch.org/databank/dailynumber/?NumberID=1331
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News Sources
sample statistic  z  SE
*
0.64  2.575  0.01
0.64  0.026
 0.614,0.666
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Confidence Interval Formula
From N(0,1)
sample statistic  z  SE
*
From original
data
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From
bootstrap
distribution
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First Born Children
• Are first born children actually smarter?
• Based on data from last semester’s class
survey, we’ll test whether first born children
score significantly higher on the SAT
X first born  X not first born  30.26
• From a randomization distribution, we find
SE = 37
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First Born Children
X first born  X not first born  30.26, SE  37
What normal distribution should we use
to find the p-value?
a)
b)
c)
d)
N(30.26, 37)
N(37, 30.26)
N(0, 37)
N(0, 30.26)
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Because this is a
hypothesis test, we want
to see what would happen
if the null were true, so
the distribution should be
centered around the null.
The variability is equal to
the standard error.
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Hypothesis Testing
Distribution of Statistic Assuming Null
Observed
Statistic
p-value
-3
-2
-1
0
1
2
3
Statistic
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p-values
If the randomization distribution is
normal:
To calculate a p-value, we just need to
find the area in the appropriate tail(s)
beyond the observed statistic of the
distribution
N(null value, SE)
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First Born Children
N(0, 37)
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p-value =
0.207
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First Born Children
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Standard Normal
• Sometimes, it is easier to just use one
normal distribution to do inference
• The standard normal distribution is
Distribution of Statistic Assuming Null
the normal distribution with mean 0 and
standard deviation 1
N  0,1
-3
-2
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-1
0
Statistic
1
2
3
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Standardized Test Statistic
•
The standardized test statistic is the
number of standard errors a statistic is
from the null value
sample statistic  null value
z
SE
•
The standardized test statistic (also
called a z-statistic) is compared to N(0,1)
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p-value
1) Find the standardized test statistic:
sample statistic  null value
z
SE
2) The p-value is the area in the tail(s)
beyond z for a standard normal
distribution
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First Born Children
1) Find the standardized test statistic
sample statistic  null value
z
SE
30.26  0

37
 0.818
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First Born Children
2) Find the area in the tail(s) beyond z
for a standard normal distribution
p-value =
0.207
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z-statistic
sample statistic  null value
z
SE
If z = –3, using  = 0.05 we would
(a) Reject the null
(b) Not reject the null
(c) Impossible to tell
(d) I have no idea
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About 95% of z-statistics are
within -2 and +2, so
anything beyond those values
will be in the most extreme
5%, or equivalently will give a
p-value less than 0.05.
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z-statistic
•
Calculating the number of standard
errors a statistic is from the null value
allows us to assess extremity on a
common scale
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Formula for p-values
From original
data
From H0
sample statistic  null value
z
SE
From
randomization
distribution
Compare z to N(0,1) for p-value
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Standard Error
• Wouldn’t it be nice if we could compute
the standard error without doing
thousands of simulations?
• We can!!!
• Or rather, we’ll be able to next week!
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To Do
 Do Project 1 (due 10/23)
 Read Chapter 5
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