Two surgical procedures are widely used to treat a certain type of cancer. To compare the success rates of thetwo procedures, random samples of the two types of surgical patients were obtained and the numbers ofpatients who showed no recurrence of the disease after a 1-year period were recorded. The data are shown inthe table. How large a sample would be necessary in order to estimate the difference in the true success ratesto within $\begin{array}{lcc}
\hline & \mathrm{n} & \text { Number of Successes } \
\hline \text { Procedure A } & 100 & 73 \
\text { Procedure B } & 100 & 92 \
\hline
\end{array}$
A) $\mathrm{n}_{1}=\mathrm{n}_{2}=104$
B) $\mathrm{n}_{1}=\mathrm{n}_{2}=73$
C) $\mathrm { n } _ { 1 } = \mathrm { n } _ { 2 } = 53$
D) $\mathrm { n } _ { 1 } = \mathrm { n } _ { 2 } = 192$ 0.10 with 95% reliability?

Question

Two surgical procedures are widely used to treat a certain type of cancer. To compare the success rates of thetwo procedures, random samples of the two types of surgical patients were obtained and the numbers ofpatients who showed no recurrence of the disease after a 1-year period were recorded. The data are shown inthe table. How large a sample would be necessary in order to estimate the difference in the true success ratesto within $\begin{array}{lcc}
\hline & \mathrm{n} & \text { Number of Successes } \
\hline \text { Procedure A } & 100 & 73 \
\text { Procedure B } & 100 & 92 \
\hline
\end{array}$
A) $\mathrm{n}_{1}=\mathrm{n}_{2}=104$ 
B) $\mathrm{n}_{1}=\mathrm{n}_{2}=73$ 
C) $\mathrm { n } _ { 1 } = \mathrm { n } _ { 2 } = 53$
D) $\mathrm { n } _ { 1 } = \mathrm { n } _ { 2 } = 192$ 0.10 with 95% reliability?

Quizplus · Accepted Answer

To estimate the difference in the true success rates to within 0.10 with 95% reliability, we use the formula for the sample size in comparing two proportions: $n = \left( \frac{Z^2 \cdot (p_1(1-p_1) + p_2(1-p_2))}{E^2} \right)$, where $Z$ is the Z-value for the desired confidence level (1.96 for 95% confidence), $p_1$ and $p_2$ are the sample proportions of success for procedures A and B respectively, and $E$ is the desired margin of error (0.10 in this case). Given the proportions of success are 73/100 for procedure A and 92/100 for procedure B, plugging these values into the formula gives a sample size of approximately 104 for each group to achieve the desired precision and confidence level.

Two Surgical Procedures Are Widely Used to Treat a Certain