SOLUTION: The following table shows the distribution of the responses of 100 employees based on two variables; gender (male or female) and opinion (in favor or againgst paying high salaries)

Question 1202484: The following table shows the distribution of the responses of 100 employees based on two variables; gender (male or female) and opinion (in favor or againgst paying high salaries).
in favor A against B Total
male 14 46 60
female 5 35 40
19 81 100
are events "female" and "in favor A" independent?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52932) (Show Source): You can put this solution on YOUR website!
The following table shows the distribution of the responses of 100 employees based on two variables;
gender (male or female) and opinion (in favor or againgst paying high salaries).

in favor A against B Total male 14 46 60 female 5 35 40 19 81 100 are events "female" and "in favor A" independent?
~~~~~~~~~~~~~~~~~~~~

To answer this question, you should check if P(female)*P(in favor A) = P(female AND in favor A). From the table, you have P(female) = = ; P(in favor A) = . P(female AND in favor A) = . Now compare if equals .

Fully explained.

Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Answer: No, the events are NOT independent.
The events are dependent somehow.

================================================================

Explanation:

Data table

in favor against total
male 14 46 60
female 5 35 40
total 19 81 100

Define these two events
F = a female employee is selected
H = the person is in favor of high salaries

There are at least two methods we can use to check if events F and H are independent or not.

---------------------------------

Method 1

If we can show P(F and H) = P(F)*P(H) is the case, then events F and H are independent.

According to the table
P(F) = (40 women)/(100 employees) = 40/100 = 2/5
P(H) = (19 in favor)/(100 employees) = 19/100
Then
P(F)*P(H) = (2/5)*(19/100) = 38/500 = 19/250

In the table there are 5 women in favor of higher salaries. Therefore P(F and H) = 5/100 = 1/20

Are the two fractions 19/250 and 1/20 equal?
Let's cross multiply to find out
19/250 = 1/20
19*20 = 250*1
380 = 250
The last statement is false, which must mean the first statement is false. The fractions 19/250 and 1/20 are not equivalent.

We could also use a calculator to get:
19/250 = 0.076
1/20 = 0.05
To help better see the results aren't the same.

Therefore P(F and H) = P(F)*P(H) is false.

Events F and H are NOT independent.
We consider them dependent.
F depends on H, or H depends on F.
The two events are linked somehow.

---------------------------------

Method 2

Events F and H are independent if and only if:
P(F given H) = P(F)
P(H given F) = P(H)

If the events were independent, then the prior "given" shouldn't affect the probability.

As previously mentioned, back in method 1, the table says:
P(F) = 2/5
P(H) = 19/100

Let's compute P(F given H).
Because we know event H has occurred, we know 100% the person is in favor of high salaries.
That means we'll narrow our focus to the "in favor" column only.

There are 5 women who favor high salaries out of 19 people total who favor high salaries.
P(F given H) = 5/19
An alternative is to use this formula
P(F given H) = P(F and H)/P(H)
I'll let the student do this part.

Compare these two items
P(F) = 2/5
P(F given H) = 5/19
Those fractions are not equal (use the cross multiplication technique, or a calculator)
This means P(F given H) = P(F) is false. The two events are dependent.

Through similar logic,
P(H given F) = 5/40 = 1/8
P(H) = 19/100

Use the technique shown in the previous method to conclude 19/100 = 1/8 is false, so P(H given F) = P(H) is also false.

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