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To test the effectiveness of the drug zopiclone for treating insomnia, critical trial was conducted on 16 (n) subjects. The mean wake time of subjects before treatment is 102.8 min; and after treatment,it is 98.2 min $\left(\stackrel{炉}{x}\right)$with standard deviation of 42.3 min (s).

It is assumed that values appear to be normally distributed.

Further, assume that the samples are selected randomly with unknown population standard deviation.

Thus, the t-distribution would be used here.

The formula for $\left(1-伪\right)\%$ confidence interval is $\stackrel{炉}{x}-E<渭<\stackrel{炉}{x}+E$.

Here, E is margin of error which is given by, $E=t_{\frac{伪}{2}}脳\frac{s}{\sqrt{n}}$

Where, $t_{\frac{伪}{2}}$ is the critical value with level of significance.

Here, $\stackrel{炉}{x}$represents the sample mean of wake time of subjects after treatment with zopiclone and represents the population mean of wake time of subjects after treatment with zopiclone.

98% level of confidence implies $伪=0.02$.

The degree of freedom is,

$$\begin{gathered} \text{df}=n-1 \\ =16-1 \\ =15 \end{gathered}$$

In the t-distribution table, find the value corresponding to the row value of degree of freedom 15 and column value of area in one tail 0.01 is 2.602 which is critical value${\text{t}}_{\text{0.01}}$.

Therefore, the critical value${\text{t}}_{\text{0.01}}$is 2.602.

Margin of error is given by,

$$\begin{gathered} E=t_{\frac{伪}{2}}脳\frac{s}{\sqrt{n}} \\ =2.602脳\frac{42.3}{\sqrt{16}} \\ =27.5212 \end{gathered}$$

Therefore, the margin of error is 27.52.

The 98% confidence interval for mean wake time after the treatment is,

$$\begin{gathered} \stackrel{炉}{x}-E<渭<\stackrel{炉}{x}+E=98.9-27.52<渭<98.9+27.52 \\ =71.38<渭<126.42 \\ 鈮�71.4<渭<126.4 \end{gathered}$$

Therefore, 98% confidence interval is $71.4\min <渭<126.4\min$.

The 98% confidence interval of the mean wake time for a population after treatment with the drug zopiclone is $71.4\min <渭<126.4\min$.

Here, the confidence interval includes mean wake time of 102.8 minbefore the treatment which means the mean wake time after the treatment with zopiclone is not different. Therefore, the treatment with the drug zopiclone does not have significant effect.