Page 18-41
Inferences concerning one proportion
Suppose that we want to test the null hypothesis, H
0
:
p = p
0
, where p
represents the probability of obtaining a successful outcome in any given
repetition of a Bernoulli trial. To test the hypothesis, we perform n repetitions
of the experiment, and find that k successful outcomes are recorded. Thus, an
estimate of p is given by p’ = k/n.
The variance for the sample will be estimated as s
p
2
= p’(1-p’)/n = k
⋅
(n-k)/n
3
.
Assume that the Z score, Z = (p-p
0
)/s
p
, follows the standard normal
distribution, i.e., Z ~ N(0,1). The particular value of the statistic to test is z
0
=
(p’-p
0
)/s
p
.
Instead of using the P-value as a criterion to accept or not accept the
hypothesis, we will use the comparison between the critical value of z0 and
the value of z corresponding to
α
or
α
/2.
Two-tailed test
If using a two-tailed test we will find the value of z
α
/2
, from
Pr[Z> z
α
/2
] = 1-
Φ
(z
α
/2
) =
α
/2, or
Φ
(z
α
/2
) = 1-
α
/2,
where
Φ
(z) is the cumulative distribution function (CDF) of the standard normal
distribution (see Chapter 17).
Reject the null hypothesis, H
0
, if z
0
>z
α
/2
, or if z
0
< - z
α
/2
.
In other words, the rejection region is R = { |z
0
| > z
α
/2
}, while the
acceptance region is A = {|z
0
| < z
α
/2
}.
One-tailed test
If using a one-tailed test we will find the value of S
, from
Pr[Z> z
α
] = 1-
Φ
(z
α
) =
α
, or
Φ
(z
α
) = 1-
α
,
Reject the null hypothesis, H
0
, if z
0
>z
α
, and H
1
: p>p
0
, or if z
0
< - z
α
, and H
1
:
p<p
0
.
