Given:
alpha level = 0.001
Population 1
success = 122
sample size n₁ = 202
Population 2
success = 220
sample size n₂ = 340
Find: test statistic and p-value
Solution:
Based on the given sample, we have two proportions here. Thus, we will be using test of two proportions formula.
[tex]z=\frac{p_1-p_2}{\sqrt[]{p(1-p)(\frac{1}{n_1}+\frac{1}{n_2})}}[/tex]
To be able to use the formula, we need to identify first the value of p, p₁, and p₂.
[tex]p=\frac{x_1+x_2}{n_1+n_2}=\frac{122+220}{202+340}=\frac{342}{542}=0.630996[/tex][tex]\begin{gathered} p_1=\frac{x_1}{n_1}=\frac{122}{202}=0.60396 \\ p_2=\frac{x_2}{n_2}=\frac{220}{340}=0.6470588 \end{gathered}[/tex]
We now have the values of p, p₁, and p₂ as well as n₁ and n₂ (sample size).
p = 0.630996
p₁ = 0.60396
p₂ = 0.6470588
n₁ = 202
n₂ = 340
Let's plug this in to the test of two proportions formula above.
[tex]z=\frac{0.60396-0.6470588}{\sqrt[]{0.630996(1-0.630996)(\frac{1}{202}+\frac{1}{340})}}[/tex]
Solve for z.
[tex]z=\frac{-0.0430988}{\sqrt[]{0.23284(\frac{271}{34340})}}=\frac{-0.0430988}{0.042866}\approx-1.005[/tex]
The test-statistic is equal to -1.005.
The p-value equivalent for this is 0.1573.
