# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### Solution 22

Since the confidence interval is $\, \bar X \pm s_{\bar X} \,$, thus by comparing it with $\, \bar X \pm z(\alpha/2)s_{\bar X} \,$, it follows that $\, z(\alpha/2) =1 \,$.

Thus we can compute $\, \alpha \,$ as:

\, \begin{align*} &\Rightarrow z(\alpha/2) = 1 \\ &\Rightarrow \Phi^{-1}(1-\alpha/2) = 1 && \text{Using the definition of z} \\ &\Rightarrow \alpha = 2(1-\Phi(1)) \\ &\Rightarrow \alpha = 0.3174 \end{align*} \,

Thus the size of confidence interval is $\, 1-\alpha = 0.6826 \,$.

$$\tag*{\blacksquare}$$