Revised August 12, 2004
Chapter 2
The Dichotomous Problem (
1
)
Introduction
The data in this chapter is of the dichotomous type of independent repeated Bernoulli
trials having constant probability of success p.
Thus, we have an experiment that has
two possible outcomes
, “success
2
” and “failure”
(called a Bernoulli trial). This experiment is repeated n times, each repetition is
independent of the others and the probability of “success” is assumed to be equal to p
and it does not change from trial to trial.
The random variable
3
of interest is
B = the number of “success”s in n independent
repetitions of this Bernoulli trial.
These are summarized in the following assumptions
Assumptions:
A1. The experiment has only
two possible outcomes
, called “success” and “failure.”
[Bernoulli experiment.]
A2. Probability of “success,” denoted by p, remains constant from trial to trial.
A3. The
n repetitions
of the experiment [trials] are
independen
t of each other.
A4.
The random variable
of interest is
B = number of “success”s in n independent trials.
2.1 A binomial Test
The null hypothesis is
Ho: p = p
0
,
where p
0
is some specified number, 0 < p
0
< 1.
The
level of significance of the test = α
= P(Type I Error) is fixed at
α
.
We have three types of alternative hypotheses (i.e., tests):
a)
OneSided, UpperTail Test,
Ha: p > p
0
b)
OneSided LowerTail Test,
Ha: p < p
0
c)
TwoSided Test
Ha: p ≠ p
0
1
Students are strongly urged to review their previous knowledge of statistical inference (point and
interval estimation as well as hypothesis testing). Also, a quick reading of chapter 1 is strongly suggested
to get the reasons for learning nonparametric (or distribution free) statistical inference techniques, its
differences from what you already (should) have learned as well as an overview of the text.
2
In this context, “success” may have nothing to do with success as used in our daily lives, it is simply
observing the outcome of the experiment that we are interested in.
3
In your previous courses you may have used X, Y, Z as symbols for the random variables. Here B is
used as the name of the random variable to remind you that the random variable has the
B
inomial
distribution.
Stat 4502 Chap 2, Page 1 of 7
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These are summarized in Table – 1.
Table – 1.
A Brief Summary of the Binomial Test
Type of Test
Hypotheses to be tested
Decision Rule
(Rejection Region)
Remarks
UpperTail
H
0
: p = p
0
vs. H
A
: p > p
0
Reject H
0
if B
≥
b
α
See Footnote
4
LowerTail
H
0
: p = p
0
vs. H
A
: p < p
0
Reject H
0
if B
≤
c
α
See Footnote
5
TwoSided
H
0
: p = p
0
vs. H
A
: p
≠
p
0
Reject H
0
if
B
≥
1
b
α
or
B
≤
2
c
α
See Footnote
6
Large Sample Approximation
Her we will use the normal approximation to the binomial distribution, which states that
when n
×
p
0
≥
10
AND n
×
(1 – p
0
)
≥
10
7
, B has an approximate normal
distribution
with mean n
×
p
0
and standard deviation
0
0
(1
)
n
p
p
×
×

(
8
). Hence
we may use the tables of the standard normal distribution for testing the hypotheses
after standardizing B, as,
*
0
0
(1
)
o
B
n
B
np
p

=

. Here B
*
has the standard normal
distribution
9
, B
*
~ N(0,1). Under these conditions, the above summary table changes to
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 Summer '11
 YESILCAY
 Bernoulli, Normal Distribution, Probability, Binomial distribution, Princeton University

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