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It is given that 39 smokers who have under gone the nicotine patch therapy are smoking after one year of the treatment, while 32 are not smoking one year after the treatment. It is claimed that most of the smokers who take the treatment have started smoking after one year of the treatment.

The null hypothesis is written as follows.

The proportion of smokers who start smoking after one year of the treatment is equal to 0.5.

$H_0:p=0.5$

The alternative hypothesis is written as follows.

The proportion of smokers who start smoking after one year of the treatment is more than 0.5.

$H_1:p>0.5$

The test is right-tailed.

The sample size is calculated below.

$$\begin{gathered} n=39+32 \\ =71 \end{gathered}$$

The sample proportion of smokers who start smoking after one year of the treatment is computed below.

$$\begin{gathered} \hat{p}=\frac{39}{71} \\ =0.549 \end{gathered}$$

The population proportion of smokers who start smoking after one year of the treatment is equal to 0.5.

The value of the test statistic is computed below.

$$\begin{gathered} z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \\ =\frac{0.549-0.5}{\sqrt{\frac{0.5\left(1-0.5\right)}{71}}} \\ =0.826 \end{gathered}$$

Thus, z=0.826.

Referring to the standard normal table, the critical value of z at for a right-tailed test is equal to 1.645.

Referring to the standard normal table, the p-value for the test statistic value of 0.826 is equal to 0.2044.

As the p-value is less than 0.05, the null hypothesis is not rejected.

There is not enough evidence to support the claim that the proportion of smokers who start smoking after one year of the treatment is greater than 50%.

Although the claim is rejected, it can be seen that the sample percentage of smokers who start smoking after one year of the treatment is equal to 54.9% (greater than 50%).

Thus, it can be said that nicotine patch therapy is not very effective in helping smokers quit smoking.