In given exercise use either the critical-value approach or the P-value approach to perform a chi square independence lest. provided the conditions for using the test are met.

Siskel In the classic TV show Sneak Previews, originally hosted by the late Gene Siskel and Roger Ebert, the two Chicago movie critics reviewed the weeks new movie releases and

then rated them thumbs up (positive), mixed, thumbs down (negative). These two critics often saw the merits of a movie differently. In general, however, were ratings given by Siskel and Ebert

associated? The answer to this question was the focus of the paper "Evaluating Agreement and Disagreement Among Movie Reviewers" by A. Agresti and L. Winner that appeared in Chance

(Vol- 10(2), pp-10-14). The following contingency table summarizes ratings by Siskel and Ebert for 160 movies.

At the 1% significance level, do the data provide sufficient evidence to conclude that an association exists between ratings of Siskel and Ebert?

$H_0$: The ratings given by Siskel and Ebert are not associated.

$H_a$: The ratings given by Siskel and Ebert are associated.

The following contingency table contains the expected frequencies corresponding to the

Observed frequencies

${(O-E)}^2$

•All expected frequencies are greater than 1, we can verify by referring to the above table.

• At most 20% of the expected frequencies are less than 5. In fact, none of the expected frequencies are less than 5, as we can verify by referring to the above table.

All assumptions are satisfied.

The test is to be performed at the 1% significance level, so$\alpha =0.01$.

$\raisebox{1ex}{${(O-E)}^2$}\!\left/ \!\raisebox{-1ex}{$E$}\right.$

the value of test statistic is${\mathcal{X}}^2=45.357$

the row variable has 3 values and column variable has 3 values, therefore r = 3, c = 3 and

$$\begin{gathered} {\mathcal{X}}^2=45.357df \\ =(r-1).(c-1) \\ =(3-1).(3-1) \\ =2\times 2 \end{gathered}$$

For, $\alpha =0.01$, Table VII reveals that the critical value is =4.

${{\mathcal{X}}_{0.01}}^2=13.277$

we see the test statistic falls in the rejection region. Thus we reject the test results are statistically significant at 1% level.

At the 1% significance level, the data provide sufficient evidence to conclude that the ratings associated by Siskel and Ebert are associated.