deviation Standard or variance?
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Standard deviation or variance?
Standard deviation has the units of
the original measurement.
e.g. mph versus mph2
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How to look at magnitudes?
Is $1 difference important?
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How to look at magnitudes?
Is $1 difference important?
Buying a new car, versus cost of gas per
gallon
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Coefficient of variation
Sometimes, helpful to look
at % deviations and
compute standard
deviation of % deviation.
i.e. standard deviation
divided by the mean
e.g. New car $1/$20,000,
very small,
e.g. Cost of gas $1/$3 =
0.33, very significant
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Statistical Inference
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Why Inference?
If we could measure population
parameters directly, why obtain a
sample of the population and estimate
sample statistics?
E.g. if we wanted to know how many
miles people drive/commute per
day/wk/yr, why not ask everyone
directly?
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Reason for Using Inference
For example, we ask people about their
income, family/marriage status,
employment, etc for the U.S. census.
Cost/effort of U.S. census for 2000,
281M people, $4.5 Billion or $16/person.
Accessibility remote areas, e.g. rural
parts of Appalachia; unwillingness;
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Sampling distribution of the sample
mean
How close is the sample mean to the
population mean? Or the variance?
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Example of the importance of the
sampling distribution
Due to recent support of alternative fuels,
an oil co. wants to find out whether
inclusion of ethanol-blended fuel actually
helps to improve sales.
Company will only continue to include this
fuel if mean daily sales are at least
18,000 gallons per station.
A random sample of 100 stations
produced mean sales of 18,200 gallons.
Should the oil co. continue w/ new blend?
(Will return to this e.g. later.)
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Using multiple dies to illustrate
sampling distribution
Roll 2 dies 20 times.
Record the average of each roll.
Compute the average of the
averages over 20 rolls.
Compute the variance of the
averages over 20 rolls
Do the same for 4 dies
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Questions:
Is the mean of x equal to the mean of x ? i.e. x ?
Is the variance of x equal to the variance of x ? i.e. x ?
2
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Trial
n=2
n=2 ave
n=4
n=4 ave
1
15
3.75
8
4
10
2.5
3
7
3.5
14
3.5
4
7
3.5
10
2.5
5
8
4
10
2.5
6
10
5
12
3
7
6
3
12
3
8
11
5.5
8
2
9
9
4.5
15
3.75
10
7
3.5
15
3.75
11
7
3.5
18
4.5
12
6
3
8
2
13
6
3
16
4
14
2
1
18
4.5
15
7
3.5
12
3
16
10
5
18
4.5
17
5
2.5
18
4.5
18
9
4.5
20
5
19
5
2.5
8
2
20
stdevp
5.5
2
mean
11
10
5
15
3.75
3.775
3.4
1.236875
0.8525
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Theoretically
x
x2 2 / n
where n is the number of dies used in a single roll.
Using equation for expected value, e.g. n=1
E ( x ) x p( x )
2 x p( x )
2
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Theoretically
x
x2 2 / n
where n is the number of dies used in a single roll.
Using equation for expected value, e.g. n=1
1 1
1
E ( x) x p( x) 1 2 ... 6 3.5
6 6
6
1
6
1
6
1
6
2 x p( x) 1 3.52 2 3.52 ... 6 3.52 2.92
2
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n=2
p( x ) is results from the various combinations that lead to each value of x
E ( x ) x p( x )
2 x x p( x )
2
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n=2
p( x ) is results from the various combinations that lead to each value of x
e.g. sample
1,1
1,2; 2,1
1,3; 2,2;3,1
x
1.0
1.5
2.0
1
2
1
E ( x ) x p( x ) 1 1.5 ... 6 3.5
36
36
36
1
2 2
2 1
1.5 3.5 ... 6 3.5 1.46
36
36
36
2 x x p( x ) 1 3.52
2
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Sampling Distributions, n=1
P(x)
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1
2
3
4
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6
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Sampling Distributions, n=2
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Sampling Distributions, n=4
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Sampling Distribution, n=30,000
Ripplestat 2010
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Summary of our experiment?
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Summary
Expected value of the mean =
expected value
Variance of the mean = variance/n
So as n increases, the variance
decreases proportionally and the
distribution looks narrower
Even if x is non-normal, mean of x
is approximately normal for large
samples
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Central Limit Theorem
Anyone heard of this?
Gaussianwave,
2010
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Central Limit Theorem
If a random sample is drawn from
any population, the sampling
distribution of the sample mean is
approximately normal for a
sufficiently large sample size.
Sometimes n>30 is considered
large, even though extremely nonnormal populations (e.g. bi-model,
skewed) be could a problem.
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Standardized Normal Distribution
What are the 2 parameters for the
normal distribution?
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Standardized Normal Distribution
The normal distribution has two parameters: mean and standard deviation .
1
f ( x)
2
e
x 2
2 2
When is this equation used?
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Standardized Normal Distribution
The normal distribution has two parameters: mean and standard deviation .
1
f ( x)
2
e
x 2
2 2
When is this equation used?
By practitioners, almost never.
This is fairly painful to integrate so we
use
a spreadsheet or a table.
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Standardized normal variable
Why do we standardize the normal variable?
i.e. subtract the mean and divide by the standard
deviation.
Z
X
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Standardized normal variable
Why do we standardize the normal variable?
i.e. subtract the mean and divide by the standard
deviation.
So that we can use a single normal table
based on a single parameter, Z, instead of
multiple tables
Z
X
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Standard Normal Table
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0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0.080
0.090
0.000
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036
0.100
0.040
0.044
0.048
0.052
0.056
0.060
0.064
0.068
0.071
0.075
0.200
0.079
0.083
0.087
0.091
0.095
0.099
0.103
0.106
0.110
0.114
0.300
0.118
0.122
0.126
0.129
0.133
0.137
0.141
0.144
0.148
0.152
0.400
0.155
0.159
0.163
0.166
0.170
0.174
0.177
0.181
0.184
0.188
0.500
0.192
0.195
0.199
0.202
0.205
0.209
0.212
0.216
0.219
0.222
0.600
0.226
0.229
0.232
0.236
0.239
0.242
0.245
0.249
0.252
0.255
0.700
0.258
0.261
0.264
0.267
0.270
0.273
0.276
0.279
0.282
0.285
0.800
0.288
0.291
0.294
0.297
0.300
0.302
0.305
0.308
0.311
0.313
0.900
0.316
0.319
0.321
0.324
0.326
0.329
0.332
0.334
0.337
0.339
1.000
0.341
0.344
0.346
0.349
0.351
0.353
0.355
0.358
0.360
0.362
1.100
0.364
0.367
0.369
0.371
0.373
0.375
0.377
0.379
0.381
0.383
1.200
0.385
0.387
0.389
0.391
0.393
0.394
0.396
0.398
0.400
0.402
1.300
0.403
0.405
0.407
0.408
0.410
0.412
0.413
0.415
0.416
0.418
1.400
0.419
0.421
0.422
0.424
0.425
0.427
0.428
0.429
0.431
0.432
1.500
0.433
0.435
0.436
0.437
0.438
0.439
0.441
0.442
0.443
0.444
1.600
0.445
0.446
0.447
0.448
0.450
0.451
0.452
0.453
0.454
0.455
1.700
0.455
0.456
0.457
0.458
0.459
0.460
0.461
0.462
0.463
0.463
1.800
0.464
0.465
0.466 0Transportation 0.468 0.469
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0.469
0.470
0.471
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Standard Normal Table
The table above (from statsoft) is the
standard normal table for positive Z
values.
i.e. shows the probability for a particular Z
value
Since the normal distribution is
symmetric, what is the probability of the
negative 1/2 of the normal curve?
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Standard Normal Table
Since the normal distribution is
symmetric, what is the probability of the
negative 1/2 of the normal curve?
0.50.
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Example: Use of Normal Table
If a probability of 95% is desired,
what is the corresponding z value?
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Example: Use of Normal Table
If a probability of 95% is desired,
what is the corresponding z value?
0.05 is 0.45 under the curve, in the
table z=1.64
A higher precision table is between
0.4495 and 0.4505 just average the
Z value, so 1.645
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Hypothesis Testing
Hypothesis testing is a tool in statistical
inference.
It is often used in transportation safety to
answer the question,
Has the safety countermeasure(s) made the
location safer?
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Hypothesis testing has 4 components
1.
2.
3.
4.
null hypothesis
alternate hypothesis
test statistic
rejection region
Follows these steps!
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1. null hypothesis
Specifies a single value for the
population parameter.
Example?
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1. null hypothesis
Specifies a single value for the
population parameter.
Ho: mean=10 crashes/year
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2. alternate hypothesis
The alternate hypothesis is usually the
question we are trying to answer.
It can assume 3 possible forms, e.g.:
a. Ha: >10 crashes/year
b. Ha: 10 crashes/year
c. Ha: <10 crashes/year
So which one (a, b or c) are we interested in
typically?
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3. test statistic
A sample statistic (a descriptive
measure) is used to reject or not
reject Ho.
Note: We do not say accept Ho.
This is usually a point estimator,
e.g. the sample mean x
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4. rejection region
If the test statistic falls within this region,
then we reject Ho.
Can someone
put this picture
into words?
Bainbridge 2010
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