9.3 Comparing two independent population proportions -- rrc math

The problem asks for a difference in proportions, making it a test of two proportions.

Let A and B be the subscripts for medication A and medication B, respectively. Then p _A and p _B are the desired population proportions.

Random variable:

H ₀ : p _A = p _B

p _A – p _B = 0

H _a : p _A ≠ p _B

p _A – p _B ≠ 0

The words "is a difference" tell you the test is two-tailed.

Distribution for the test: Since this is a test of two binomial population proportions, the distribution is normal:

$p_c=\frac{x_A+x_B}{n_A+n_B}=\frac{20+12}{200+200}=0.081–p_c=0.92$

${P^{\prime }}_A–{P^{\prime }}_B~N\left[0,\sqrt{(0.08)(0.92)(\frac{1}{200}+\frac{1}{200})}\right]$

P′ _A – P′ _B follows an approximate normal distribution.

Calculate the p -value using the normal distribution: p -value = 0.1404.

Estimated proportion for group A: ${p^{\prime }}_A=\frac{x_A}{n_A}=\frac{20}{200}=0.1$

Estimated proportion for group B: ${p^{\prime }}_B=\frac{x_B}{n_B}=\frac{12}{200}=0.06$

Graph:

P′ _A – P′ _B = 0.1 – 0.06 = 0.04.

Half the p -value is below –0.04, and half is above 0.04.

Compare α and the p -value: α = 0.01 and the p -value = 0.1404. α < p -value.

Make a decision: Since α < p -value, do not reject H ₀ .

Conclusion: At a 1% level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference in the proportions of adult patients who did not react after 30 minutes to medication A and medication B .