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Section 5.1
Normal
Distributions
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Outline
 Density curves
 Normal distribution
 Finding normal probabilities (technology)
 Finding normal endpoints (technology)
 Standard normal
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Some 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)
40
Dot Plot
60
0.2
0.0
r
-0.2
-0.4
-0.6
What do Diff means: Finger taps
Mean :Body TemperaturesAll bell-shaped
distributions
you notice?
Measures from Sample of BodyTemp50
98.2
98.3
98.4
0.6
0.4
Dot Plot
Measures from Scrambled CaffeineTaps
98.5
98.6
Nullxbar
98.7
98.8
Proportion : Owners/dogs
Measures from Sample of Collection 1
98.9
0.3 Unlocking
0.4
0.5
0.6
0.7 Data
0.8
Statistics:
the Power
of
phat
99.0
-4
Dot Plot
Dot Plot
-3
-2
-1
0
Diff
1
2
3
Mean : Atlanta commutes
Measures from Sample of CommuteAtlanta
26
27
28
29
xbar
30
4
Dot Plot
31
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Density Curve
A density curve is a theoretical model
to describe a variable’s distribution.
Think of a density curve as an idealized
histogram, where:
(1) The total area under the curve is one.
(2) The proportion of the population in any
interval is the area over that interval.
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Density for Bootstrap Means
for Atlanta Commutes
What proportion are
between 30 and 31?
Area is
about 0.15
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Normal Distribution
A normal distribution has a symmetric
bell-shaped density curve.
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Parameters of a Normal
Two features distinguish one normal density
from another:
• The mean is its center of symmetry (μ).
• The standard deviation controls its spread (σ).
Notation:
X~N(μ,σ)
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N(µ,σ)
σ
µ2σ µσ
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σ
μ
µ+σ µ+2σ
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Example: A Population
Verbal SAT ~ N( 580, 70)
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Example: Bootstrap Distribution
Original
sample
𝑥’s for Atlanta commutes ≈ N( 29.11, 0.93)
Bootstrap std. dev. (SE)
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Ex: Randomization Distribution
H0
𝑝’s for dog/owners matches ≈ N( 0.5, .10)
Randomization std. dev. (SE)
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How can we find areas under a
normal density?
N(μ,σ)
We need
technology!
a
Calculus!
b
b
Area  a
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1
e
2 
( x )2

2 2
dx
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StatKey
Pick the tail
Adjust μ, σ
Probability
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Endpoint
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Example: Verbal SAT scores
Suppose that verbal SAT scores for applicants at
a college follow a normal distribution with
mean µ = 580 and std. dev. σ =70.
What proportion of applicants have SAT scores
above 650?
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Example: Verbal SAT scores
About 4.3%
of applicants
will have
verbal SAT
scores above
700
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Example: Bootstrap Means
Suppose that the bootstrap distribution of
means for samples of size 500 Atlanta commute
times is N(29.11,0.93).
Find an endpoint (percentile) so that just 5% of
the bootstrap means are smaller.
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Example: Bootstrap Means
About 5% of
the bootstrap
means will be
less than
27.58 minutes.
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Note: All that really matters is the number of
Finding
Probabilities
for N(μ,σ)
standard
deviations
from the mean.
About what proportion should be within one
std. dev. of the mean?
≈68%
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Standard Normal
=0, =1  Z~N(0,1)
To convert any X~N(μ,σ) to Z~N(0,1):
𝑋−𝜇
𝑍=
𝜎
(z-score)
“Standardize” the endpoint(s), then use Z~N(0,1)
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Ex: Dog/Owner randomization
proportions, 𝑝 ≈ N(0.5,0.1)
?
Original 𝑝
0.64
X-area above 0.64= Z-area
0.64−0.5
above
0.1
= 1.40
area = 0.081
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Converting Normals
𝑋−𝜇
𝑍=
𝜎
X~N(μ, σ)
Z~N(0,1)
𝑋 = 𝜇 + 𝑍𝜎
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Example: Percentile for Verbal SAT
25%-tile (Q1) for
Z~N(0,1) is -0.674.
Find the 25%-tile for the
Verbal SAT~N(580,70)
distribution
X= μ + Zσ = 580+(-0.674)(70) = 533
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GOALS
If we can approximate a bootstrap
distribution with a normal …
… construct a confidence interval.
If we can approximate a randomization
distribution with a normal …
… compute a p-value.
IF we can find an easy way to estimate SE, we can
even do this without generating the distribution!
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What is the area below -1.50 in a
standard normal distribution?
A. 0.067
B. 0.933
C. 0.241
D. 0.759
E. 0.500
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What is the area above 2.20 in a
standard normal distribution?
A. 0.003
B. 0.997
C. 0.014
D. 0.986
E. 0.037
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What is the area between 0.8 and 1.4
in a standard normal distribution?
A. 0.247
B. 0.185
C. 0.028
D. 0.476
E. 0.131
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What is the endpoint z in a standard
normal distribution if the area to the
right of z is 0.03 ?
A. 0.247
B. 1.881
C. 0.897
D. 2.158
E. 1.751
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What is the endpoint z in a standard
normal distribution if the area to the
left of z is 0.18?
A. 0.915
B. -1.254
C. 1.762
D. -2.158
E. -0.915
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What is the endpoint z in a standard
normal distribution if the area
between z and –z is 0.60?
A. 0.200
B. 0.842
C. 1.168
D. -2.158
E. 0.400
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Summary
 Statistical inference is drawing conclusions about a
population based on a sample
 We use a sample statistic to estimate a population
parameter
 To assess the uncertainty of a statistic, we need to
know how much it varies from sample to sample
 To create a sampling distribution, take many samples
of the same size from the population, and compute
the statistic for each
 Standard error is the standard deviation of a statistic
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