## Exam 1

1. #### Question

The daily expenses of summer tourists in Vienna are analyzed. A survey with $132$ tourists is conducted. This shows that the tourists spend on average $168.8$ EUR. The sample variance ${s}_{n-1}^{2}$ is equal to $230.6$.
Determine a $95%$ confidence interval for the average daily expenses (in EUR) of a tourist.

1. What is the lower confidence bound?
2. What is the upper confidence bound?

#### Solution

The $95%$ confidence interval for the average expenses $\mu$ is given by:
 $\begin{array}{ccc}\multicolumn{1}{c}{}& \hfill & \left[\stackrel{‾}{y} - 1.96\sqrt{\frac{{s}_{n-1}^{2}}{n}},\mathrm{ }\stackrel{‾}{y} + 1.96\sqrt{\frac{{s}_{n-1}^{2}}{n}}\right]\hfill \\ \multicolumn{1}{c}{}& =\hfill & \left[168.8 - 1.96\sqrt{\frac{230.6}{132}},\mathrm{ }168.8 + 1.96\sqrt{\frac{230.6}{132}}\right]\hfill \\ \multicolumn{1}{c}{}& =\hfill & \left[166.209, 171.391\right].\hfill \end{array}$

1. The lower confidence bound is $166.209$.
2. The upper confidence bound is $171.391$.