Question

Vasectomies and Prostate Cancer. In the United States, approximately 450,000 vasectomies are performed each year. 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. $269\left(7\right)$, 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? Consider men who had had a vasectomy Population $2.$

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.

Short answer

a) At $1\%$ level of significance, the data provide sufficient evidence to reject the null hypothesis.

b) This is an observational study.

c) There could not be another variable that might be related to both.

Step-by-step solution

Part(a) Step 1: Given Information

The given values are,

$$\begin{gathered} x_1=69, \\ {n}_1=21300, \\ {x}_2=113, \\ {n}_2=22000. \end{gathered}$$

Part(a) Step 2: Explanation

The formula for ${\tilde{p}}_1$is given by,

${\tilde{p}}_1=\frac{x_1}{n_1}$

The formula for ${\tilde{p}}_2$is given by,

${\tilde{p}}_2=\frac{x_2}{n_2}$

The formula for $z$is given by,

$Z_{\text{calculated}}=\frac{{\tilde{p}}_1-{\tilde{p}}_2}{\sqrt{{\tilde{p}}_p\left(1-{\tilde{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

The formula for ${\tilde{p}}_p$is given by,

${\tilde{p}}_p=\frac{x_1+x_2}{n_1+n_2}$

The value of ${\tilde{p}}_1$is calculated as,

$$\begin{gathered} {\tilde{p}}_1=\frac{x_1}{n_1} \\ =\frac{69}{21300} \\ =0.003 \end{gathered}$$

The value of ${\tilde{p}}_2$is calculated as,

$$\begin{gathered} {\tilde{p}}_2=\frac{x_2}{n_2} \\ =\frac{113}{22000} \end{gathered}$$

$=0.005$

The value of ${\tilde{p}}_p$is calculated as,

$$\begin{gathered} {\tilde{p}}_p=\frac{x_1+x_2}{n_1+n_2} \\ =\frac{69+113}{21300+22000} \\ =0.004203 \\ \tilde{q}=1-{\tilde{p}}_p \\ =1-0.0042 \\ =0.9958 \end{gathered}$$

The value of $z$is calculated as,

$Z_{\text{calculated}}=\frac{{\tilde{p}}_1-{\tilde{p}}_2}{\sqrt{{\tilde{p}}_p\left(1-{\tilde{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

$=\frac{0.003239-0.005136}{\sqrt{0.0042(1-0.0042)}\left(\sqrt{\frac{1}{21300}+\frac{1}{22000}}\right)}$

$$\begin{gathered} =\frac{-0.001897}{0.0006219} \\ =-3.05 \end{gathered}$$

Calculate$p-$value is.

$$\begin{gathered} p-\text{value}=P\left(Z<Z_{\text{calculated}}\right) \\ =P(Z<-3.05) \\ =(=NORMSDIST(-3.05\left)\right) \\ =0.001144 \end{gathered}$$

If the $p$- value is less than the given significance level, we reject the null hypothesis and say we have enough evidence to conclude that $p1$ is less than $p2$ at the $1\%$ level of significance.

As a result, at a 1% level of significance, the data provide enough evidence to reject the null hypothesis.

Part(b) Step 1: Given Information

To find: Whether this is an observational study or designed experiment study.

The given values are,

$$\begin{gathered} x_1=69, \\ {n}_1=21300, \\ {x}_2=113, \\ {n}_2=22000 \end{gathered}$$

Part(b) Step 2: Explanation

Using the previously solved solution in the preceding section.

We are conducting an observational study because we are not randomly assigning people to one of two groups.

As a result, this is an observational study.

Part(c) Given Information

The given values are,

$$\begin{gathered} x_1=69, \\ {n}_1=21300, \\ {x}_2=113, \\ {n}_2=22000 \end{gathered}$$

Part(c) Step 2: Explanation

Using the previously solved solution in the preceding section.

We cannot conclude this based on the evidence presented above because there could be another variable that is related to both.

As a result, there could not be another variable that is related to both.