Answer in Statistics and Probability for THEBURBURGUY #248070
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.
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,
“displaystylesum_{i=1}^3r_i=displaystylesum_{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=displaystylesum_{i=1}^3r_i=displaystylesum_{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(2space decimalspace places)”
“E_{21}=E_{22}=E_{23}=(270*200)/600=90”
“E_{31}=E_{32}=E_{33}=(112*200)/600=37.33(2space decimalspace places)”
The chi-square test statistic is given as,
“chi_c^2=displaystylesum_{i=1}^3displaystylesum_{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+n(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(3space decimalspace 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.
