Exam 1

1. Question

   The daily expenses of summer tourists in Vienna are analyzed. A survey with $71$ tourists is conducted. This shows that the tourists spend on average $130$ EUR. The sample variance $s_{n-1}^2$ is equal to $83.2$.
   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 $\mathit{\mu }$ is given by:
   

   ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{}& \hfill & [\bar{y}\hspace{0.5em}-\hspace{0.5em}1.96\sqrt{\frac{s_{n-1}^2}{n}},\mathrm{\hspace{0.5em}\hspace{0.5em}}\bar{y}\hspace{0.5em}+\hspace{0.5em}1.96\sqrt{\frac{s_{n-1}^2}{n}}]\hfill \\ \multicolumn{1}{c}{}& =\hfill & [130\hspace{0.5em}-\hspace{0.5em}1.96\sqrt{\frac{83.2}{71}},\mathrm{\hspace{0.5em}\hspace{0.5em}}130\hspace{0.5em}+\hspace{0.5em}1.96\sqrt{\frac{83.2}{71}}]\hfill \\ \multicolumn{1}{c}{}& =\hfill & [127.878,\hspace{0.5em}132.122].\hfill \end{array}}$

   
   
   
   
   1. The lower confidence bound is $127.878$.
   2. The upper confidence bound is $132.122$.