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\(\begin{array}{l}{\rm{Sample size of first sample: }}{n_1} = 56{\rm{ and }}{x_1} = 18\\{\rm{Sample size of second sample:}}{n_2} = 51{\rm{ and }}{x_2} = 12\\{\rm{Level of significance }}\alpha = 0.10\end{array}\)

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis and the alternative hypothesis state the opposite of each other. The null hypothesis needs to contain an equality.

\(\begin{array}{l}{H_0}:{p_1} = {p_2}\\{H_a}:{p_1} \ne {p_2}\end{array}\)

\({\rm{ The sample proportion is the number of successes divided by the sample size: }}\)

\(\begin{array}{c}{{\hat p}_1} = \frac{{{x_1}}}{{{n_1}}} = \frac{{18}}{{56}} \approx 0.3214\\{{\hat p}_2} = \frac{{{x_2}}}{{{n_2}}} = \frac{{12}}{{51}} \approx 0.2353\\{{\hat p}_p} = \frac{{{x_1} + {x_2}}}{{{n_1} + {n_2}}}\\ = \frac{{18 + 12}}{{56 + 51}}\\ = \frac{{30}}{{107}}\\ \approx 0.2804\end{array}\)

Determine the value of the test statistic:

\(\begin{array}{c}z = \frac{{{{\hat p}_1} - {{\hat p}_2}}}{{\sqrt {{{\hat p}_p}\left( {1 - {{\hat p}_p}} \right)} \sqrt {\frac{1}{{{n_1}}} + \frac{1}{{{n_2}}}} }}\\ = \frac{{0.3214 - 0.2353}}{{\sqrt {0.2804(1 - 0.2804)} \sqrt {\frac{1}{{56}} + \frac{1}{{51}}} }}\\ \approx 0.99\end{array}\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, assuming that the null hypothesis is true. Determine the P-value using table the normal probability table in the appendix:

\(\begin{array}{c}P = P(Z < - 0.99{\rm{ or }}Z > 0.99)\\ = 2P(Z < - 0.99)\\ = 2(0.1611)\\ = 0.3222\end{array}\)

If the P-value is smaller than the significance level, then reject the null hypothesis:

\(P > 0.10 \Rightarrow {\rm{ Fail to reject }}{H_0}\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme, assuming that the null hypothesis is true. Determine the P-value using table the normal probability table in the appendix:

\(\begin{array}{c}P = P(Z < - 0.99{\rm{ or }}Z > 0.99)\\ = 2P(Z < - 0.99)\\ = 2(0.1611)\\ = 0.3222\end{array}\)

If the P-value is smaller than the significance level, then reject the null hypothesis:

\(P > 0.10 \Rightarrow {\rm{ Fail to reject }}{H_0}\)

There is not sufficient evidence to support the claim that the true proportion of TC individuals who recidivate during the specified follow-un period differs from the proportion of DJS individuals who do so.