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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
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asked 2021-03-06
Use either the critical-value approach or the P-value approach to perform the required hypothesis test. Approximately 450,000 vasectomies are performed each year in the United States. In this surgical procedure for contraception, the tube carrying sperm from the testicles is cut and tied. Several studies have been conducted to analyze the relationship between vasectomies and prostate cancer. The results of one such study by E. Giovannucci et al. appeared in the paper “A Retrospective Cohort Study of Vasectomy and Prostate Cancer in U.S. Men” (Journal of the American Medical Association, Vol. 269(7), pp. 878-882). Of 21,300 men who had not had a vasectomy, 69 were found to have prostate cancer, of 22,000 men who had had a vasectomy, 113 were found to have prostate cancer. a. At the 1% significance level, do the data provide sufficient evidence to conclude that men who have had a vasectomy are at greater risk of having prostate cancer? b. Is this study a designed experiment or an observational study? Explain your answer. c. In view of your answers to parts (a) and (b), could you reasonably conclude that having a vasectomy causes an increased risk of prostate cancer? Explain your answer.

Answers (1)

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

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