Question

The head supervisor of three discount pharmacy stores wants to know how many discount coupons are redeemed in his stores. The findings of the survey are listed in the table below. At α=0.01

α=0.01, test the hypothesis that redemption level and location are related. 

                             

Answer 1

The hypothesis tested here is,

$H_0:$ Redemption level and location are statistically independent

Against

$H_1$ :Redemption level and location are statistically dependent.

To conduct this test, the chi-square test of independence is used as described below.

Let;

$E_{ij}$ be the expected cell count for the observed cell count in row $i$ and column $j$.

$r_i$ be the total count for row $i$

$c_j$ be the total count for column $j$ and

$n$ be the total count given by,

$\displaystyle\sum_{i=1}^3r_i=\displaystyle\sum_{j=1}^3c_j=n$

For the data above,

$r_1=218$

$r_2=270$

$r_3=112$ and

$c_1=c_2=c_3=200$

$n=\displaystyle\sum_{i=1}^3r_i=\displaystyle\sum_{j=1}^3c_j=600$

First, the expected cell counts are computed using the formula below,

$E_{ij}=(r_i*c_j)/n$

Therefore,

$E_{11}=E_{12}=E_{13}=(218*200)/600=72.67(2\space decimal\space places)$

$E_{21}=E_{22}=E_{23}=(270*200)/600=90$

$E_{31}=E_{32}=E_{33}=(112*200)/600=37.33(2\space decimal\space places)$

The chi-square test statistic is given as,

$\chi_c^2=\displaystyle\sum_{i=1}^3\displaystyle\sum_{j=1}^3(O_{ij}-E_{ij})^2/E_{ij}$

$\chi_{c}^2=(69-72.67)^2/72.67+(97-72.67)^2/72.67+(52-72.67)^2/72.67+(101-90)^2/90+ (93-90)^2/90+(76-90)^2/90+(30-37.33)^2/37.33+(10-37.33)^2/37.33+(72-37.33)^2/37.33=71.476(3\space decimal\space places)$

This statistic is compared with the chi-square table value with $(r-1)(c-1)=(3-1)(3-1)=2*2=4$ degrees of freedom at $\alpha=0.01$ significance level where $r$ is the total number of rows and $c$ is the total number of columns.

It is given as,

$\chi_{0.01,4}^2=13.2767$

Since the test statistic$(\chi_{c}^2=71.476)$ is greater than the table value$(\chi_{c}^2=13.2767)$, we reject the null hypothesis and conclude that sufficient evidence exist to show that the redemption level and location are related.