1 The following data comes from the 2011 Behavioral Risk Fac

1. The following data comes from the 2011 Behavioral Risk Factor Surveillance System (BRFSS) survey, which is run by the Centers for Disease Control (CDC). This is a joint probability table for the proportions of survey respondents who smoke and who have had heart attacks. Answer the following questions: (a) What is the proportion of people who smoke? (b) What is the proportion of people who have had a heart attack? (c) If a person is a smoker, are they more likely to have had a heart attack than someone who is not a smoker? Hint: Be careful what this question is asking! (4) If someone tells you that they have had a heart attack, what is the probability that they are also a smoker? (e) Is smoking independent of having a heart attack? (f) Define Bernoulli random variables. S = 0 (non-smoker), S = 1 (smoker), and H = 0 (no heart attack), H = 1 (heart attack). What is Cov(S, H)? (g) What is p(S, H)?

Solution

(a) the proportion of people who smoke = 0.03+ 0.44 = 0.47

(b)  the proportion of people who have had a heart attack= 0.03+0.03 =0.06

(c) p ( heart attack | he is smoker )= 0.03 / 0.47=0.0638

p ( heart attack | non-smoker ) = 0.03 / 0.53 = 0.0566
so, If a person is a smoker, they are more likely to have had a heart attack than someone who is not a smoker....

d) p ( he is smoker | he have had a heart attack) = 0.03 / 0.06 = 0.5

e) p( smoking and heart attack) = 0.03 which is not equals to p(smoker)*p(heart attack).....so,they are not independent...

f) E(S) = P(S=1) = 0.47

E(H) =P(H=1)= 0.06
E(SH)=P(S=1 AND H=1) = 0.03

cov(S,H) = 0.03 - ( 0.47*0.06) = 0.0018..
G) (S, H) = 0.0018 / [ sqrt( 0.06*0.94) * sqrt( 0.47*0.53) ] = 0.015185