Question

(1 point) Recall in the course we learnt how to display data of two categorical variables by means of a contingency table. In this question, we will formulate a measure to quantify the level of association between the two categorical variables. Such a measure is often used in a statistical test called Chi-square test for assessing whether there is an association between two categorical variables. This question is also used to motivate the learning of independence and to connect the concept back to what we have learnt in the Course How is diet type (high cholesterol diet versus low cholesterol diet) related to the risk of coronary heart disease? Data of 23 individuals: No heart diseaseTotal Heart disease (i) 11 (iii) 4 High cholesterol diet 15 Low cholesterol diet (ii) 2 (iv) 6 13 10 23 From the table we find that the probability of having heart disease is 13/23 and the probability of having high cholesterol diet is 15/23. Similarly, we can find the probability of not having heart disease and the probability of having low cholesterol diet. Part a If there is no association between the two variables (i.e., the two are independent), the probability of having heart disease and high cholesterol diet is: [Round to four decimal places] Part b If the two variables are independent, we should expect the number of individuals with heart disease and high cholestoral diet to be the probability in Part a multiplied by 23 individuals, which is: [Round to two decimal places]

Answer 1

Part a:

If there is no association between the two variables:

P(Heart disease AND High cholesterol) = P(Heart disease) X (P(High cholesterol)

                                                          = 13/23 X 15/23

                                                            = 0.5652 X 0.6522

                                                           = 0.3686

So,

Answer is:

0.3686

Part b:

Expected Number of Heart disease AND High cholesterol = 13 X 15/23 = 8.48

So,

Answer is:

8.48

Part c:

From the given data, Expected Frequencies are got as follows:

	Heart disease	No heart disease	Total
High cholesterol diet	13 X 15/23 = 8.48	10 X 15/13 = 6.52	15
Low cholesterol diet	13 X 8/23 = 4.52	10 X 8/23 = 3.48	8
total	13	10	23

Test Statistic $\chi ^{2}$ is got as follows:

Observed (O)	Expected (E)	(O - E)2/E
11	8.48	0.75
4	6.52	0.98
2	4.52	1.41
6	3.48	1.83
Total = $\chi ^{2}$ =		4.96

So,

Answer is:

$\chi ^{2}$ = 4.96