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Inferences concerning one proportion
Suppose that we want to test the null hypothesis, H
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
Assume that the Z score, Z = (p-p
distribution, i.e., Z ~ N(0,1). The particular value of the statistic to test is z
(p'-p
)/s
.
0
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
Pr[Z> z
] = 1-Φ(z
α
/2
where Φ(z) is the cumulative distribution function (CDF) of the standard normal
distribution (see Chapter 17).
Reject the null hypothesis, H
In other words, the rejection region is R = { |z
acceptance region is A = {|z
One-tailed test
If using a one-tailed test we will find the value of S , from
Pr[Z> z
Reject the null hypothesis, H
p<p
.
0
)/s
, follows the standard normal
0
p
) = α/2, or Φ(z
α
/2
, if z
>z
, or if z
< - z
α
0
0
/2
0
| < z
}.
α
0
/2
) = α, or Φ(z
] = 1-Φ(z
α
α
, if z
>z
, and H
: p>p
α
0
0
1
: p = p
, where p
0
0
2
= p'(1-p')/n = k⋅(n-k)/n
p
, from
α
/2
) = 1- α/2,
α
/2
.
α
/2
| > z
}, while the
α
0
/2
) = 1- α,
α
, or if z
< - z
, and H
α
0
0
Page 18-41
3
.
=
0
:
1
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