Question

A survey of 1,000 men and women asked, "Do you earn over $55,000 per year?" The table below shows the responses for males and females.

Answer 1

Male, Female, Total
Income over $55,000 475, 375, 850
Income below $55,000 75, 75, 150
Total 550, 450, 1,000

Based on these data, are "being male" and "earning over $55,000" independent events? (5 points)


A. No, P(being male | the person earns over $55,000) ≠ P(being male)
B. No, P(being male | the person earns over $55,000) = P(being male)
C. Yes, P(being male | the person earns over $55,000) = P(being male)
D. Yes, P(being male | the person earns over $55,000) ≠ P(being male)

Answer 2

I think independence = true if
P(A|B) = P(A) <--- or that A does not depend on B
and if not true, then I guess we have not independent

and I think
P(A | B) = P(A) * P(B)

if so then... let
P(A) = P(person earns over $55k)
P(B) = P(male)

Then ask is this true,
is P(person earns over $55k) * P(male) = P(male) ???
if so then we have independence.

But
Is P(male) = P(550 /1000) ???
or
is P(male) in this case considered to be P(475/850) ?
and
is P(Person earns over 55k) = P(850/1000) ?

Answer 3

Have you calculated the probability of a person being male, given that the person earns over $55,000?

Answer 4

so P(475/850)

Answer 5

and do we then compare that with just the probability of being male?
P(550/1000) ?

Answer 6

Correct. So the probability of a person being male, given that the person earns over $55,000 = 0.5588.
The probability of a person being male is 0.550.
When we look at the choices, the only correct one is A. Do you agree?

Answer 7

looks good to me.. thank you.
confusing, that my use of "P(male)" is ambiguous in a sense..
I must have the set notation confused I think.. will research it a bit and think about it.

thank you

Answer 8

You're welcome :)

Answer 9

There are two tests for independence:
\[P(A \cap B)=P(A) \times P(B)\]
Also this test:
\[P(A|B)=P(A)\]
The second test is used for your question. As you can see the events in question are not independent

Answer 10

So if I extract the numbers from the table

P(A | B) is constructed as 475/850
The B part is taken from the total for the row, and the A part is taken from the male column.

and means the probability of A given B, or probability of male given income of 55k.

then the P(A) is constructed from the male column total / subject total, meaning probability of male given total number of people.

Answer 11

Good work! You have understood :)