Question

A survey conducted by a leading HMO found that of 2000 women, 340 were heavy smokers and 25 had emphysema. Of those who had emphysema, 21 were also heavy smokers. Using the data in this survey, determine whether the events being a heavy smoker and having emphysema are independent?

Answer 1

The events being a heavy smoker and having emphysema are not independent.

Explanation:

We are given the following in the question:

Number of women = 2000

Number of heavy smoker = 340

Number of women who has emphysema = 25

Number of women who has emphysema and are heavy smoker = 21

`\text{Probability} = \displaystyle\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}`

P(Smoker) =

`P(S) = (340)/(2000)=0.17`

P(emphysema) =

`P(E) = (25)/(2000)= 0.0125`

P(Smoker and emphysema) =

`P(S\cap E) = (21)/(2000) = 0.0105`

Two events A and B are said to be independent if

`P(A\cap B)=P(A)* P(B)`

Checking conditions for independence:

`P(S)* P(E) = 0.17* 0.0125=0.002125\\P(S)* P(E) \\eq 0.0105\\P(S)* P(E)\\eq P(S\cap E)`

Thus, the events being a heavy smoker and having emphysema are not independent.