Testing The Difference Between Two Proportions, - HP 49g+ User Manual

Testing the difference between two proportions
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 for two populations 1 and 2.
hypothesis, we perform n
and find that k successful outcomes are recorded. Also, we find k
1
outcomes out of n trials in sample 2. Thus, estimates of p
2
respectively, by p ' = k
1 1
The variances for the samples will be estimated, respectively, as
2
s = p '(1-p ')/n = k
1 1 1 1
And the variance of the difference of proportions is estimated from: s
2
s .
2
Assume that the Z score, Z = (p
distribution, i.e., Z ~ N(0,1). The particular value of the statistic to test is z
(p '-p '-p )/s .
1 2 0 p
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.
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 z
repetitions of the experiment from population 1,
1
/n , and p ' = k /n .
1 2 2 2
⋅(n 3 2
-k )/n , and s = p
1 1 1 1 2
-p -p )/s , follows the standard normal
1 2 0 p
) = α/2, or Φ(z
α
/2
, if z >z , or if z
α
0 0 /2 0
| < z }.
α
0 /2
: p -p = p , where the p's
0 1 2 0
To test the
successful
2
and p are given,
1 2
⋅(n
'(1-p ')/n = k -k )/n
2 2 2 2 2 2
2
= s
p
, from
α
/2
) = 1- α/2,
α
/2
< - z .
α
/2
| > z }, while the
α
0 /2
, from
a
Page 18-42
3
.
2
2
+
1
=
0