# Testing The Difference Between Two Proportions - HP F2226A - 48GII Graphic Calculator User Manual

Graphing calculator.

## 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