# Mathematical Statistics and Data Analysis - Solutions

### Chapter 7, Survey Sampling

#### Solution 19

For the first part, we need to find the constant $\, k \,$ such that the interval $\, (-\infty, \bar X + ks_{\bar X}) \,$ is a 90% confidence interval for $\, \mu \,$.

We shall use the same function here $\, z(\alpha) \,$ as used in the book. Also let $\, Z \,$ denotes the standard normal function.

Thus we have:

Thus for 90% confidence interval we need $\, \alpha = 0.9 \,$. And $\, k = z(1-\alpha) \,$. By the definition of $\, z \,$, we have $\, z(\alpha) = \Phi^{-1}(1-\alpha) \,$. Thus $\, k= \Phi^{-1}(1-(1-\alpha)) = \Phi^{-1}(0.9) \,$

Now we just need to lookup in the standard normal table to find $\, k = 1.28 \,$.

Second part is also similar where we need to find the value of $\, k \,$ for the one sided confidence interval $\, (\bar X - ks_{\bar X}) \,$.

We can proceed from the last result:

Thus for 95% confidence interval we need $\, \alpha = 1-0.95=0.05 \,$. And $\, k = z(\alpha) \,$. By the definition of $\, z \,$, we have $\, z(\alpha) = \Phi^{-1}(1-\alpha) \,$. Thus $\, k= \Phi^{-1}(1-\alpha) = \Phi^{-1}(0.95) = 1.645 \,$.

$$\tag*{\blacksquare}$$