Question # Use either the critical-value approach or the P-value approach to perform the required hypothesis test. Approximately 450,000 vasectomies are performe

Study design
ANSWERED  2021-03-07

Step 1 a) $$\displaystyle{H}_{{{0}}}:{p}_{{{1}}}={p}_{{{2}}}$$
$$\displaystyle{h}_{{{a}}}:{p}_{{{1}}}{<}{p}_{{{2}}}$$ The sample proportion is the number of successes divided by the sample size: $$\displaystyle{w}{i}{d}{e}\ \hat{{{p}}}_{{{1}}}={\frac{{{x}_{{{1}}}}}{{{n}_{{{1}}}}}}={\frac{{{69}}}{{{21300}}}}\approx{0.003}$$
$$\displaystyle{w}{i}{d}{e}\ \hat{{{p}}}_{{{2}}}={\frac{{{x}_{{{2}}}}}{{{n}_{{{2}}}}}}={\frac{{{113}}}{{{22000}}}}={0.005}$$
$$\displaystyle{w}{i}{d}{e}\ \hat{{{p}}}_{{{p}}}={\frac{{{x}_{{{1}}}+{x}_{{{2}}}}}{{{n}_{{{1}}}+{n}_{{{2}}}}}}={\frac{{{69}+{113}}}{{{21300}+{22000}}}}={\frac{{{182}}}{{{43300}}}}={0.004}$$ Determine the value of the test statistic: $$z=\frac{\widehat{p}_{1}-\widehat{p}_{2}}{\sqrt{\widehat{p}_{p}(1-\widehat{p}_{p})}\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}}=\frac{0.003-0.005}{\sqrt{0.004(1-0.004)\sqrt{\frac{1}{21300}+\frac{1}{22000}}}}\approx -3.30$$ Determine the P-value using table 2: $$\displaystyle{P}={0.0005}$$ If the P-value is smaller than the significance level, reject the null hypothesis: $$\displaystyle{P}{<}{0.01}={1}\%\Rightarrow$$ Reject $$\displaystyle{H}_{{{0}}}$$

Step 2 b) An experiment deliberately imposes some trearment on individuals in order to observe their responses. An observational study tries to gather information without disturbing the scene they are observing. Observational study c) Yes, because we rejected the null hypothesis in (a). Answer: a. Yes b. Observational study c. Yes