Answer to Question #272554 in Statistics and Probability for Glen Lincon

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. [3] 

Answer 1

i)

The hypotheses to be tested in this question are,

$H_0:$ Gender of the person and their opinion of the quality of the restroom facilities are independent.

$Against$

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

ii)

The critical region for this test is given as $\chi^2_{\alpha, df}$ , where $\alpha=0.01$ is the desired level of significance and $df$ are the degrees of freedom given as,

$df=(r-1)(c-1)$ where $r$ and $c$ are the number of rows and columns respectively.

For this case, there are $r=3$ rows and $c=2$ columns. Therefore $df=(3-1)(2-1)=2$ and the critical value is $\chi^2_{0.01,2}=9.21034$

iii)

The expected value for Male and below average is given as

$E=(41*35)/100=14.35$ where 41 is the total for the males column and 35 is the total for the below average row while 100 is the sample size.

Thus, the expected value for Male and below average is 14.35.

iv)

For this test the null hypothesis is rejected if the test statistic $\chi^2_*$ is greater than the critical value $\chi^2_{0.01,2}$.

A test statistic of $\chi^2_*=9.825\gt\chi^2{0.01,2}=9.21034$ and therefore, the null hypothesis is rejected and we conclude that there is no sufficient evidence to show that gender of the person and their opinion of the quality of the restroom facilities are independent at 1% level of significance.