Download supplement10 - Ka

Supplement 10:
The Power of a Test
*The ppt is a joint effort: Mr DAI Liang discussed the power of a test with
Dr. Ka-fu Wong on 16 April 2007; Ka-fu explained the problem; Liang
drafted the ppt; Ka-fu revised it. Use it at your own risks. Comments, if
any, should be sent to kafuwong@econ.hku.hk.
Imagine this scenario



At a starry night, we noticed a bright spot in the sky.
From the map, we knew that it is a system of binary stars. However,
our naked eyes could not tell.
So, we borrowed a telescope and looked at it again.
If it is actually two stars…


Given the same telescope (or naked eyes), the larger the distance
between the two stars, the more likely we can distinguish between
them.
Given the same two stars, the larger the Magnification Coefficient
(MC) of our telescope, the more likely we can distinguish between
the two stars.
The idea of “power”!!



The power a test is a measure of the ability of a test in
distinguishing between two possible values of the parameter of
interest.
In the context of hypothesis testing, the power of the test is the
ability of the test in telling us the null is wrong when the true
parameter is different from the null.
Conditional (given) a test, such ability depends on how far apart the
null is from the true parameter.
 The larger the difference between the truth and the null, the
larger the power of a given test.
Analogy between statistical test and
telescope
Statistical test
Telescope (Star gazing)
Power of a test in distinguishing
Magnification coefficient (MC) of
between two values of parameter. your telescope
The larger the difference between The larger the distance between
the truth and the null, the larger
the two stars, the more likely we
the power.
can distinguish between them
using the same telescope.
Given the same truth and null,
the larger is the power, the more
accurate is the test.
Given the same two stars, the
larger the MC of our telescope,
the more likely we can distinguish
the two stars.
Exercise (Problem 10.42)

A random sample of 802 supermarket shoppers had 378 shoppers
that preferred generic brand items if the price was lower.
 Test at the 10% level the null hypothesis that at least one-half
of all shoppers preferred generic brand items against the
alternative that the population proportion is less than one-half.

Find the power of a 10% level test if, in fact, 45% of the
supermarket shoppers are able to state the correct price of an
item immediately after putting it into the cart.
The null hypothesis



The hypotheses:
 Null hypothesis (H0): p  0.5,
 Alternative (H1): p < 0.5
Variance of the sample proportion p under H0 is:
 0.5*(1-0.5)/802=0.000312
Level of significance = 0.1
When is H0 rejected?




Reject H0 if sample proportion p is too small.
At the level of significance (=0.1), z = -1.28
Upper limit of rejection is 0.5+ z*[std. dev. under H0] = 0.4774
Therefore, H0 is rejected when p  0.4774
Rejection region
=0.1
0.4774
-1.28
p=378/802 =0.4713
Null rejected.
p=0.5
p
m=0
z
Standardized to standard normal: z=(p-p)/std(p)
If the real proportion is 0.45





H0 is false since p = 0.45 ≠ 0.5
The power is Pr(rejecting the null | p=0.45)
= Pr(p  0.4774 | p=0.45)
Variance under p = 0.45 is :
 .45*(1-.45)/802=.000309
Pr(p  0.4774 | p=0.45)
=Pr(Z  (.4774 - .45)/sqrt(.000309))
=0.9404
Thus, 0.9404 is the probability that H0 (p  0.5) is correctly rejected
when the truth is p = 0.45.
Supplement 9:
An Example of Switching Null and
Alternative Hypothesis
A simulated example of a binary star, where two bodies with similar mass orbit around a
common barycenter in elliptic orbits. (From http://en.wikipedia.org/wiki/Binary_star)
- END -