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In a sample of 270 subjects who were given a placebo, 7 developed allergic reactions. In another sample of 863 subjects who were treated with the drug, 8 developed allergic reactions.

The sample proportion of subjects who were given a placebo and developed allergic reactions is computed below:

$$\begin{gathered} {\hat{p}}_1=\frac{7}{270} \\ =0.0259 \end{gathered}$$

Hence,

$$\begin{gathered} {\hat{q}}_1=1-{\hat{p}}_1 \\ =1-0.0259 \\ =0.9741 \end{gathered}$$

The sample proportion of subjects who were treated with the drug and developed allergic reactions is computed below:

$$\begin{gathered} {\hat{p}}_2=\frac{8}{863} \\ =0.0093 \end{gathered}$$

Hence,

$$\begin{gathered} {\hat{q}}_2=1-{\hat{p}}_2 \\ =1-0.0093 \\ =0.9907 \end{gathered}$$

The given level of significance is 0.05.

Therefore, the value of$z_{\frac{伪}{2}}$ from the standard normal table is equal to 1.96.

The margin of error corresponding to the placebo group is computed below:

$$\begin{gathered} E_1=z_{\frac{伪}{2}}脳\sqrt{\frac{{\hat{p}}_1{\hat{q}}_1}{n_1}} \\ =1.96脳\sqrt{\frac{0.0259脳0.9741}{270}} \\ =0.0190 \end{gathered}$$

Therefore, the margin of error corresponding to the placebo group is equal to 0.0190.

The margin of error corresponding to the treatment group is computed below:

$$\begin{gathered} E_2=z_{\frac{伪}{2}}脳\sqrt{\frac{{\hat{p}}_1{\hat{q}}_1}{n_2}} \\ =1.96脳\sqrt{\frac{0.0093脳0.9907}{863}} \\ =0.0064 \end{gathered}$$

Therefore, the margin of error corresponding to the treatment group is equal to 0.0064.

The 95% confidence interval estimate of the proportion of subjects who were given a placebo and developed allergic reactions is computed below:

$$\begin{gathered} {\hat{p}}_1-E_1<p<{\hat{p}}_1+E_1 \\ 0.0259-0.0190<p<0.0259+0.0190 \\ 0.0070<p<0.0449 \\ 0.70\%<p<4.49\% \end{gathered}$$

Thus, the 95% confidence interval estimate of the proportion of subjects who were given a placebo and developed allergic reactions is equal to (0.70%, 4.49%).

The 95% confidence interval estimate of the proportion of subjects who were treated with the drug and developed allergic reactions is computed below:

$$\begin{gathered} {\hat{p}}_2-E_2<p<{\hat{p}}_2+E_2 \\ 0.0093-0.0064<p<0.0093+0.0064 \\ 0.0029<p<0.0157 \\ 0.29\%<p<1.57\% \end{gathered}$$

Thus, the 95% confidence interval estimate of the proportion of subjects who were treated with the drug and developed allergic reactions is equal to (0.29%, 1.57%).

Upon observing the confidence intervals, it can be said that the percentage of subjects who were given a placebo and developed allergic reactions is comparatively more than the percentage of subjects who were treated with the drug and developed allergic reactions.

Therefore, there is no particular instance of the occurrence of allergic reactions associated with the drug.