1 MF-852 Financial Econometrics Lecture 4 Probability Distributions and Intro. to Hypothesis Tests...
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Transcript of 1 MF-852 Financial Econometrics Lecture 4 Probability Distributions and Intro. to Hypothesis Tests...
1
MF-852 Financial Econometrics
Lecture 4 Probability Distributions and Intro. to
Hypothesis Tests
Roy J. EpsteinFall 2003
2
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
3
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.
4
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
5
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)?
6
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.
7
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
8
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)
9
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!
10
N(0,1) and Standardized Variables Suppose x is N(12,10).
What is P(x 24.8) ?
11
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
12
13
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
14
Properties of Sample Mean
Sample mean is unbiased!
11)(
11)(
111
n
nnxE
nxE
nxE
nn
i
n
i
15
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)(
16
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)
17
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.
18
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.
19
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%.
20
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
21
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?
22
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
23
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
24
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.
25
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)
26
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?
27
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
28
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?