Question

b) Persons who visit the restroom of a certain fast-food outlet were asked to state their opinion of the quality of the restroom facilities, The following tables show the responses from a sample of 100 persons. Gender of Respondent Totals Male Female Quality of Facilities Above Average 8 7 15 Average 26 24 50 Below Average 7 28 35 Totals 41 59 100 A 𝜒 2 test is carried out to determine whether there is an association between the gender of persons and their opinion. i) State appropriate null and alternative hypotheses [2] ii) Determine the critical region of the test at the 1% level of significance [1] iii) Calculate the expected value for Male and below average [2] iv) For a test statistic of 9.825, explain with reason, the conclusion of your test. [

Answer 1

$i)$ The hypotheses for this problem are,

$H_0:$ gender of persons and their opinion on the quality of the restroom facilities are independent.

$Against$

$H_1:$ Gender of persons and their opinion on the quality of the restroom facilities are not independent.

$ii)$

To determine the critical region, we need to find the degrees of freedom as follows,

The degrees of freedom for this test are given as,

$df=(r-1)(c-1)$ where $r=3$ is the number of rows and $c=2$ is the number of columns.

Therefore, the number of degrees of freedom is $df=(3-1)(2-1)=2*1=2$. Thus, the critical region at $\alpha=0.01$ is given as,

$\chi^2_{\alpha,2}=\chi^2_{0.01,2}=9.21034$

$iii)$

The expected value for Male and below average is found using the formula below,

$E_{ij}=(r_i*c_j)/n$ where $n=100$ .

$r_i$ is the row total and $c_j$ is the column total. For this case, $i=3$ and $j=1$.

Therefore, the expected count is,

$E_{31}=(r_3*c_1)/n=(35*41)/100=14.35$

$iv)$

The test statistic $\chi^2_c=9.825$ is compared with the table value provided in $(ii)$ above and the null hypothesis is rejected if $\chi^2_c\gt \chi^2_{0.01,2}$ .

Since $\chi^2_{c}=9.825\gt \chi^2_{0.01,2}=9.21034$, we reject the null hypothesis and conclude that there is no sufficient evidence to show that gender of persons and their opinion on the quality of the restroom facilities are independent. Therefore,  there is an association between gender of persons and their opinion on the quality of the restroom facilities.