1 mf-852 financial econometrics lecture 4 probability distributions and intro. to hypothesis tests...
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![Page 1: 1 MF-852 Financial Econometrics Lecture 4 Probability Distributions and Intro. to Hypothesis Tests Roy J. Epstein Fall 2003](https://reader031.vdocument.in/reader031/viewer/2022032201/56649d395503460f94a12ad6/html5/thumbnails/1.jpg)
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MF-852 Financial Econometrics
Lecture 4 Probability Distributions and Intro. to
Hypothesis Tests
Roy J. EpsteinFall 2003
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Distribution of a Random Variable A random variable takes on
different values according to its probability distribution.
Certain distributions are especially important because they describe a wide variety of random variables.
Binomial, Normal, student’s t
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Binomial Distribution Random variable has two
outcomes, 1 (“success”) and 0 (“failure”) Coin flip: heads = 1, tails = 0 P(success) = p P(failure) = q = (1 – p)
Binomial distribution yields probability of x successes in n outcomes.
Excel will do the calculations.
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Tails and Body of a Distribution
Binomial Distributionp = 0.4, n = 8
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
0 1 2 3 4 5 6 7 8
Successes
Pro
bab
ility
upper tail
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Binomial Example (RR p. 20) Medical treatment has p = .25. n = 40 patients What is probability of at least 15
successes (cures) I.e, P(x 15)?
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Normal Distribution A normally distributed random
variable: Is symmetrically distributed around
its mean Can take on any value from – to + Has a finite variance Has the famous “bell” shape
“Standard normal:” mean 0, variance 1.
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Standard Normal Distribution
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
-3
-2.9
-2.7
-2.6
-2.4
-2.3
-2.1 -2
-1.8
-1.7
-1.5
-1.4
-1.2
-1.1
-0.9
-0.8
-0.6
-0.5
-0.3
-0.2 0
0.1
5
0.3
0.4
5
0.6
0.7
5
0.9
1.0
5
1.2
1.3
5
1.5
1.6
5
1.8
1.9
5
2.1
2.2
5
2.4
2.5
5
2.7
2.8
5 3
z
f(z)
tail area
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N(0, .5) Distribution
0.000
0.200
0.400
0.600
0.800
1.000
1.200
1.400
1.600
1.800
-3
-2.9
-2.7
-2.6
-2.4
-2.3
-2.1 -2
-1.8
-1.7
-1.5
-1.4
-1.2
-1.1
-0.9
-0.8
-0.6
-0.5
-0.3
-0.2 0
0.15 0.3
0.45 0.6
0.75 0.9
1.05 1.2
1.35 1.5
1.65 1.8
1.95 2.1
2.25 2.4
2.55 2.7
2.85 3
z
f(z)
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N(0,1) Probabilities Suppose z has a standard normal
distribution. What is: P(z 1.645)? P(z –1.96)? Excel will tell us!
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N(0,1) and Standardized Variables Suppose x is N(12,10).
What is P(x 24.8) ?
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Key Properties of Normal Distribution Sum of 2 normally distributed
random variables is also normally distributed.
The distribution of the average of independent and identically distributed NON-NORMAL random variables approaches normality. Known as the Central Limit Theorem Explains why normality is so
pervasive in data
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Sample Mean Take a sample of n independent
observations from a distribution with an unknown . Data are n random variables x1, …
xn.
We estimate the unknown population mean with the sample mean “xbar”:
n
1in
1xx
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Properties of Sample Mean
Sample mean is unbiased!
11)(
11)(
111
n
nnxE
nxE
nxE
nn
i
n
i
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Properties of Sample Mean
Sample mean has variance. But the variance is reduced with more data.
n
nnn
xVarn
xVarn
xVarnn
i
n
i
2
2
1
2
12
12
11)(
11)(
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Null Hypothesis “Null hypothesis” (H0) asserts a
particular value (0) for the unknown parameter of the distribution.
Written as H0 : = 0 E.g., H0 : = 5
H0 usually concerns a value of particular interest (e.g., given by a theory)
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Null Hypothesis xbar is unlikely to equal 0
exactly. Samples have sampling error, by
definition. Is xbar still consistent with H0
being a true statement? This involves a hypothesis test.
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Hypothesis Testing Hypothesis testing finds a range
for called the confidence interval.
The confidence interval is the set of acceptable hypotheses for , given the available data.
H0 is accepted if the confidence interval includes 0.
Otherwise H0 is rejected.
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Confidence Interval confidence interval = xbar
allowable sampling error How wide should the interval be
around xbar? Customary to use a 95%
confidence interval. The interval will include the true
95% of the time Each tail probability is 2.5%.
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Construction of Confidence Interval If x1, … xn are normally distributed
then xbar is normally distributed. Then:
The 95% confidence interval is
%95)96.1)/(
96.1( 0
n
xP
%95)96.196.1( 0 n
xn
xP
nx
96.1
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Confidence Interval Example You are a restaurant manager. Burgers
are supposed to weigh 5 ounces on average. The night shift makes burgers with a standard deviation of 0.75 ounces.
You eat 12 burgers from the night shift and xbar is 5.4 ounces. What is a 95% confidence interval for the weight of the night shift burgers?
You eat 8 more burgers that have an average weight of 5.25 ounces. What is a 95% confidence interval for this sample?
What is a 95% confidence interval based on all 20 burgers?
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Sample Variance Usually the population variance, as
well as the mean, is unknown. Estimate 2 with the sample
variance:
We divide by n-1, not n. What is the sample variance of xbar?
n
i xxn
s1
22 )(1
1
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Sample Variance Usually the population variance, as
well as the mean, is unknown. Estimate 2 with the sample
variance:
We divide by n-1, not n. What is the sample variance of xbar?
n
i xxn
s1
22 )(1
1
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t-distribution Confidence intervals use the t-
distribution instead of the normal when the variance is estimated from the sample.
T-distribution has fatter tails than the normal.
Confidence intervals are wider because we have less information.
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t distribution (3 dof)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-3
-2.9
-2.7
-2.6
-2.4
-2.3
-2.1 -2
-1.8
-1.7
-1.5
-1.4
-1.2
-1.1
-0.9
-0.8
-0.6
-0.5
-0.3
-0.2 0
0.15 0.3
0.45 0.6
0.75 0.9
1.05 1.2
1.35 1.5
1.65 1.8
1.95 2.1
2.25 2.4
2.55 2.7
2.85 3
t
f(t)
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Confidence Interval with t-distribution You hired Leslie, a new salesperson.
Leslie made the following sales each month in the first half:
January — $25,000 April — $20,000 February — $27,000 May — $22,000 March — $29,000 June —
$35,000 What is a 95% confidence interval for
Leslie’s monthly sales? (assume monthly sales are normally distributed)
Suppose you knew that the standard deviation of sales was $1,500. How would your conclusion change?
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Significance Levels Assuming H0, what is the
probability that the sample value would be as extreme as the value we actually observed? Alternative to confidence interval
Equal to
variatesnormalfor ))/(
( 0
n
xzP
atesfor t vari))/(
( 0
ns
xtP
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Type 1 and Type 2 Error Accept or reject H0 based on the
confidence interval. Type 1 error: reject H0 when it is
true. What is probability of this?
Type 2 error: accept H0 when it is false. How important is this?