Transcript Lec 6, Ch.5, pp90-105: Statistics (Objectives)

Lec 6, Ch.5, pp90-105: Statistics
(Objectives)
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Understand basic principles of statistics through reading
these pages, especially…
Know well about the normal distribution
Know the special characteristics of the Poisson
distribution
Understand the meaning of correlation and dependence
Understand what confidence intervals mean
Learn how to estimate sample sizes for data collections
Understand the concept of hypothesis testing
What we cover in class today…
Anything not covered in class, you learn them from reading pp.95-105.
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The normal distribution – how to read the
standard normal distribution table
Central limit theory (CLT)
The Poisson distribution – why it is relevant to
traffic engineering
Correlation and dependence
Confidence bounds and their implications
Estimating sample sizes
The concept of hypothesis testing
The normal distribution
Mean = 55 mph
What’s the
probability the next
value will be less than
65 mph?
From the sample
normal distribution
to the standard
normal
distribution.
z = (x - µ)/ 
= (65 – 55)/7
= 1.43
Use of the standard normal distribution
table, Tab 5-1
Z = 1.43
Most popular one is 95% within µ ± 1.96
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Central limit theorem (CLT)
Definition: The population may have any unknown distribution with
a mean µ and a finite variance of  2. Take samples of size n from
the population. As the size of n increases, the distribution of sample
means will approach a normal distribution with mean µ and a
variance of  2/n.
F(x)
f (X )

X
approaches
µ
X distribution
X ~ any (µ,  2)
x
µ
X distribution
X ~ N ( ,  X 2 )
X
The Poisson distribution (“counting
distribution” or “Random arrival”)
m xem
P( X  x) 
x!
With mean µ = m and
variance 2 = m.
If the above characteristic is
not met, the Poisson does not
apply.
 The binomial distribution tends to approach the Poisson distribution with
parameter m = np. (See Table 4-3)
 When time headways are exponentially distributed with mean  = 1/, the
number of arrivals in an interval T is Poisson distributed with mean = m =
T.
Correlation and dependence
y = f(x)
Linear regression:
y = a + bx
Non-linear regression:
y = axb (example)
Correlation coefficient
r (1, perfect fit)
Coefficient of determination
Independent variable x
r2 (Tells you how much of
variability can be “explained” by
the independent variables.)
Confidence bounds and interval
Point estimates: A point estimate is a single-values
estimate of a population parameter made from a sample.
Interval estimates: An interval estimate is a
probability statement that a population parameter is
between two computed values (bounds).
µ
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-
X
True population mean
Point estimate of X from a
sample
X
X – tas/sqrt(n)
X + tas/sqrt(n)
Two-sided interval
estimate
Confidence interval (cont)
When n gets larger (n>=30), t can become z. The
probability of any random variable being within 1.96
standard deviations of the mean is 0.95, written as:
P[(µ - 1.96)  y  (µ + 1.96)] = 0.95
Obviously we do not know µ and  . Hence we
restate this in terms of the distribution of sample
means:
P[( x - 1.96E)  y  ( x + 1.96E)] = 0.95
Where, E = s/SQRT(n)
(Review 1, 2, 3, and 4 in page 100.)
Estimating sample sizes
For cases in which the distribution of means can be
considered normal, the confidence range for 95%
confidence is:
s
 1.96
n
If this value is called the tolerance (or “precision”),
and given the symbol e, then the following equation
can be solved for n, the desired sample size:
s
e  1.96
n
and
s2
n  3.84 2
e
By replacing 1.96 with z and 3.84 with z2, we can use
this for any level of confidence.
(Review 1 and 2 on page 101.)
The concept of hypothesis testing
Two distinct choices:
Null hypothesis, H0
Alternative hypothesis: H1
E.g. Inspect 100,000 vehicles, of which 10,000 vehicles are “unsafe.”
This is the fact given to us.
H0: The vehicle being tested is “safe.”
H1: The vehicle being tested is “unsafe.”
In this inspection,
15% of the unsafe vehicles are determined to be safe Type II error (bad error)
and 5% of the safe vehicles are determined to be unsafe  Type I error
(economically bad but safety-wise it is better than Type II error.)
We want to minimize
especially Type II
error.
Types of errors
Steps of the Hypothesis Testing
Decision
Reality
Reject H0
Accept H0
H0 is true
Type I error
Correct
H1 is true
Correct
Type II error
State the hypothesis
Select the significance level
Compute sample statistics
and estimate parameters
Compute the test statistic
Fail to reject a false
null hypothesis
Reject a correct null hypothesis
Determine the acceptance
and critical region of the test
statistics
Reject or do not reject H0
P(type I error) =  (level of significance)
P(type II error ) = 
Dependence between , , and sample
size n
There is a distinct relationship between the two probability values 
and  and the sample size n for any hypothesis. The value of any one
is found by using the test statistic and set values of the other two.
 Given  and n, determine . Usually the  and n values are the
most crucial, so they are established and the value is not controlled.
 Given  and , determine n. Set up the test statistic for  and 
with H0 value and an H1 value of the parameter and two different n
values.
The t (or z) statistics is: t
or z

( X  )

n
(Use an example from a stat book)
One-sided and two-sided tests
 The significance of the hypothesis test is indicated by , the type I error
probability.  = 0.05 is most common: there is a 5% level of significance,
which means that on the average a type I error (reject a true H0) will occur 5 in
100 times that H0 and H1 are tested. In addition, there is a 95% confidence level
that the result is correct.
 If H1 involves a not-equal relation,
no direction is given, so the
significance area is equally divided
between the two tails of the testing
distribution.
 If it is known that the parameter can
go in only one direction, a one-sided
test is performed, so the significance
area is in one tail of the distribution.
0.025
each
Two-sided
0.05
One-sided upper