Download Hypothesis Testing

Hypothesis Testing
• Two siblings, Arlen and an Robin
agree to resolve their disputed
ownership of an Ert´e painting by
tossing a penny. Arlen produces a
penny and, just as Robin is about
to toss it in the air, Arlen
smoothly suggests that spinning
the penny on a table might
ensure better randomization.
Robin assents and spins the
penny. As it spins, Arlen calls
“Tails!” The penny comes to rest
with Tails facing up and Arlen
takes possession of the Ert´e.
Robin is left with the penny.
• That evening, Robin wonders if she has been
had. She decides to perform an experiment.
She spins the same penny on the same table
100 times and observes 68 Tails. It occurs to
Robin that perhaps spinning this penny was
not entirely fair, but she is reluctant to accuse
her brother of impropriety until she is
convinced that the results of her experiment
cannot be dismissed as coincidence. How
should she proceed?
• What is the true value of p? More precisely,
what is a reasonable guess as to the true value
of p?
• Is p = 0.5? Specifically, is the evidence that p
6= 0.5 so compelling that Robin can
comfortably accuse Arlen of impropriety?
• What are plausible values of p? In particular, is
there a subset of [0, 1] that Robin can
confidently claim contains the true value of p?
Another Example
A sociologist has been hired to assess the effectiveness of a
rehabilitation program for alcoholics in her city. The program
serves a large area, and she does not have the resources to
test every single client. Instead, she draws a random sample
of 127 people from the list of all clients and questions them
on a variety of issues. She notices that, on the average, the
people in her sample miss fewer days of work each year than
the city as a whole. Are alcoholics treated by the program
more reliable than workers in general?
• We can see that there is a difference in rates of
absenteeism and that the average rate of absenteeism
for the sample is lower than the rate for the
community.
• Although it’s tempting, we can’t make any conclusions
yet because we are working with a random sample of
the population we are interested in, not the population
itself (all people treated in the program).
Explanation 1
• The first explanation, which we will call explanation A,
is that the difference between the community mean of
7.2 days and the sample mean of 6.8 days reflects a
real difference in absentee rates between the
population of all treated alcoholics and the community.
• The difference is statistically significant in the sense
that it is very unlikely to have occurred by random
chance alone.
• If explanation A is true, the population of all treated
alcoholics is different from the community and the
sample did not come from a population with a mean
absentee rate of 7.2 days.
Explanation 2
• The second explanation, or explanation B, is that the
observed difference between sample and community
means was caused by mere random chance.
• In other words, there is no important difference
between treated alcoholics and the community as a
whole, and the difference between the sample mean
of 6.8 days and the mean of 7.2 days of absenteeism
for the community is trivial and due to random chance.
• If explanation B is true, the population of treated
alcoholics is just like everyone else and has a mean
absentee rate of 7.2 days.
0.4
0.3
0.2
0.1
0.0
𝜇 = 7.2
5
6
7
8
9
10
Remember that this is the distribution of 𝑥 s. From sample to sample measured
𝜎
values deviate. Also remember that the deviation is 𝑁 .
• Which explanation is correct? As long as we
are working with a sample rather than the
entire group, we cannot be absolutely (100%)
sure about the answer to this question.
• However, we can set up a decision-making
procedure so conservative that one of the two
explanations can be chosen knowing that the
probability of choosing the incorrect
explanation is very low.
This decision-making process begins with the
assumption that explanation B is correct.
Symbolically, the assumption that the mean
absentee rate for all treated alcoholics is the
same as the rate for the community as a whole
can be stated as
𝜇 = 7.2 𝑑𝑎𝑦𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟
• If explanation B (the population of treated alcoholics is
not different from the community as a whole and has a
μ of 7.2) is true, then the probability of getting the
observed sample outcome ( 𝑋 = 6.8) can be found.
• Let us add an objective decision rule in advance. If the
odds of getting the observed difference are less than
0.05 we will reject explanation B.
• If this explanation were true, a difference of this size
(7.2 days vs. 6.8 days) would be a very rare event, and
in hypothesis testing we always bet against rare
events.
𝜎 unknown
1.Read Chapter 7- Chapter 9 from Essentials of
Statistics by Joseph F. Healey
2.Read Chapter 8,9 and 10 from Research
Methods by Charles Stangor