Question

(2 points) 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 leaming of independence and to connect the concept back to what we have learnt in the course. Let's revisit the example we have looked at 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 High cholesterol diet Low cholesterol diet Heart disease No heart disease Total (1) 11 (1) 4 (1) 2 (iv) 6 10 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. Parta If there is no association between the two variables (le the two are independent), the probability of having heart disease and high cholesterol diet is [Round to four decimal places Part 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 Parta multiplied by 23 individuals, which is [Round to two decimal places Part Repeating Part we find that the expected number of individuals for the cells (). (M) () respectively on the table are: 452,6 52, 3.48 The folicwing measure (caled Ch square test statistic) Observed Expected) Expected quantifies the level of association between two categorical variables. The symbol means a sum "Observed" here refers to the observed counts on the table, while "Expected refers to the expected counts given independence for the two variables is true The sum is taken across all the cells) to (I) on the table If there is no association, the observed counts should not differ very much from the expected counts, which results in a relatively small value of A large value indicates disagreement between the expected and observed counts which suggests the assumption of independence does not hold and the two variables are likely to be associated Computex Round to two decimal places Of course, how large is large is another problem and this is beyond the scope of this course Seh

Answer 1

Part a.
If the two variables are independent, probability of having heart disease and high cholesterol diet
= probability of having heart disease * probability of having high cholesterol diet
= (13/23) * (15/23) = 0.3686

Part b.
Expect number of individuals with heart disease and high cholesterol diet = (13 * 15) / 23 = 8.48

Part c.
$\chi^2$ = (11 - 8.48)² / 8.48 + (2 - 4.52)² / 4.52 + (4 - 6.52)² / 6.52 + (6 - 3.48)² / 3.48 = 4.95