Question

Save Answer response. Question 1 of 8 Question 1 5 points The following data represent the results of a recent study of 800 people regarding the relationship between their smoking status and types of illness. The objective is to test if the types of illness is related to smoking status. Use 5% significance level. Illness No smokers smokers Total Hart problem 50 150 200 Cancer 100 500 600 total 150 650 800 We wish to test if smoking status and types illness are related. a. Calculate the test statistic (round to 2 decimals) b. Find the critical value c. What is your conclusion? Answer: Since the test statistic is (greater/less) than the critical value, we have enough evidence/ don't have enough evidence) to conclude that the types of illness and smoking status are related.

Answer 1

$\text{We can employ test of association using a contingency table.}\\ \text{Let, H,C,N,S stand for Heart problem, Cancer, Non-smokers and Smokers respectively}\\ \text{Then, the null is :}\\ H_{0}:\text{ Illness types and smoking status are indep.}\\ \text{We can use the chi sq. test for this: }\\ \text{We know that under null,}\\ T=\sum_{i\in\{H,C\};j\in\{S,N\}}\frac{(f_{ij}-\frac{f_{i.}f_{.j}}{f_{..}})^2}{\frac{f_{i.}f_{.j}}{f_{..}}}\sim \chi^2_{(2-1)(2-1)}=\chi^2_{1}\\ \text{where }f_{ij}\text{ is the frequency for ith illness category and jth smoking category}\\ f_{i.},f_{.j},f_{..}\text{ are the respective marginals and the grand total respectively}\\ \text{We reject for large values of }T\\ \text{The observed value of T is: (not showing the cumbersome calculations. Can be easily done using the expression above by substituting the values)}\\ a.\text{ } t= 6.84\\$

b.We reject for large values of T. The level of significance is 5%. the critical value is thus

C= X6.05;1 = 3.84 c. Since we observe that t>c,, we reject the null. Hence, illness and smoking status are not independent. Since the test statistic is greater than the critical value, we have enough evidence to conclude that the types of illness and smoking status are related.