Hypothesis Testing Double Proportion

Question

Students Safe Driving has targeted seat-belt usage as a positive step to reduce accidents and injuries. Before a major campaign at one college, 44 percent of 150 drivers entering the college parking lotwere using their seat belts. After the seat-belt awareness program, the proportion using seat-belts had risen to 52 percent in a sample of 200 vehicles. At a 0.04 significance level, can the students conclude that their campaign was effective?

Answer 1

We will test statistically the null-hypothesis $$H_0: p_2=p_1$$ to the alternative $$H_A:p_2>p_1$$ at significance level $α=0.04$ (one-sided test: we want to know if the percentage "increased"). Let $$p^*=\frac{n_1\hat{p_1}+n_2\hat{p_2}}{n_1+n_2}=\frac{44\%\cdot150+52\%\cdot200}{150+200}=\frac{66+104}{350}=\frac{170}{350}=0.4857$$ Test statistic $$z=\frac{\hat{p_1}-\hat{p_2}}{\sqrt{p^*(1-p^*)\left(\frac1{n_1}+\frac1{n_2}\right)}}=\frac{0.44-0.52}{\sqrt{0.486(1-0.486)\left(\frac1{150}+\frac1{200}\right)}}=-1.48$$ From a standard normal table we can calculate the p-value $$p=P(Z\le z)=\Phi(-1.48)=0.0694>0.04$$ so we do not reject the null hypothesis. So, no the students cannot conclude that their campaign was effective, at the $0.04$ significance level.