91Ó°ÊÓ

A recent debate about where in the United States skiers believe the skiing is best prompted the following survey. Test to see if the best ski area is independent of the level of the skier.

The Table with data is given by

The table with expected values:

We want to test these hypotheses:

H_$0$ : Ski area is independent of the level of the skier

H_$1$: Ski area depends on the level of the skier

Since there are $3$ rows and $3$columns, the number of degrees of freedom is $(3-1)(3-1)=4$

Test statistics is given by

${\mathrm{χ}}^2=\frac{(20−12.4{)}^2}{12.4}+\frac{(10−13.8{)}^2}{13.8}+\frac{(10−13.8{)}^2}{13.8}+$

$+\frac{(30−31{)}^2}{31}+\frac{(30−34.5{)}^2}{34.5}+\frac{(40−34.5{)}^2}{34.5}+$

$+\frac{(40−46.6{)}^2}{46.6}+\frac{(60−51.7{)}^2}{51.7}+\frac{(50−51.7{)}^2}{51.7}$

$=4.7+0.03+0.93+1.05+0.59+$

$+1.332+1.046+0.877+0.056$

$\text{Using the applet, we get that}p\text{-value is}0.03185\text{:}$

$\text{Taking}\mathrm{Î\pm }=0.05\text{, we can see that}$

$p=0.03185<0.05$

This means that we reject the null hypothesis. We conclude that the ski area and level of the skier are not independent.