A survey of 1,200 men and women asked, "Do you earn over $75,000 per year?" The table below shows the responses for males and females.

Male Female Total
Income over $75,000 585 485 1,070
Income below $75,000 65 65 130
Total 650 550 1200
Based on these data, are "being female" and "earning over $75,000" independent events? (5 points)

No, P(being female | the person earns over $75,000) = P(being female)
No, P(being female | the person earns over $75,000) ≠ P(being female)
Yes, P(being female | the person earns over $75,000) = P(being female)
Yes, P(being female | the person earns o

ver $75,000) ≠ P(being female)

To determine if "being female" and "earning over $75,000" are independent events, we need to compare the conditional probability of being female given the person earns over $75,000 (P(being female | the person earns over $75,000)) with the probability of being female (P(being female)).

We can calculate these probabilities using the given data:

P(being female | the person earns over $75,000) = Number of females earning over $75,000 / Total number of people earning over $75,000 = 485 / 1,070

P(being female) = Number of females / Total number of people = 550 / 1,200

Now we can compare these probabilities:

P(being female | the person earns over $75,000) / P(being female) = (485 / 1,070) / (550 / 1,200)

Simplifying this expression, we find:

P(being female | the person earns over $75,000) / P(being female) = (485 * 1,200) / (1,070 * 550)

Calculating this expression:

P(being female | the person earns over $75,000) / P(being female) ≈ 1.508

Since P(being female | the person earns over $75,000) is not equal to P(being female), we can conclude that "being female" and "earning over $75,000" are not independent events.

Therefore, the correct answer is:

No, P(being female | the person earns over $75,000) ≠ P(being female).

To determine if "being female" and "earning over $75,000" are independent events, we need to compare the probability of being female given that the person earns over $75,000 (P(being female | the person earns over $75,000)) to the probability of being female (P(being female)).

To calculate P(being female | the person earns over $75,000), we use the formula:

P(being female | the person earns over $75,000) = Number of females earning over $75,000 / Total number of people earning over $75,000

From the information provided in the table, we can see that the number of females earning over $75,000 is 485 and the total number of people earning over $75,000 is 1,070.

P(being female | the person earns over $75,000) = 485 / 1,070

To calculate P(being female), we use the formula:

P(being female) = Number of females / Total number of people

From the information provided in the table, we can see that the number of females is 550 and the total number of people is 1,200.

P(being female) = 550 / 1,200

Now, we can compare the two probabilities:

P(being female | the person earns over $75,000) = 485 / 1,070
P(being female) = 550 / 1,200

If P(being female | the person earns over $75,000) is equal to P(being female), then "being female" and "earning over $75,000" are independent events. If P(being female | the person earns over $75,000) is not equal to P(being female), then they are dependent events.

After calculating the probabilities, if P(being female | the person earns over $75,000) is equal to P(being female), the correct answer would be "Yes, P(being female | the person earns over $75,000) = P(being female)." However, if they are not equal, the correct answer would be "No, P(being female | the person earns over $75,000) ≠ P(being female)."

P(being female | the person earns over $75,000)

=P(F∩O)/P(O)
=(585/1200)/(1070/1200)
=585/1070

P(F)=650/1200

=>P(F∩O)/P(O) ≠ P(F),
i.e. P(F∩O) ≠ P(F)*P(O)
therefore they are not independent.