I was given a table that showed a simple random sample of adults' age and annual income living in suburb of a large city.
a) What is the probability that a person chosen at random from those in this sample will be the 31-45 age category? The answer I got is .43 or 43%.
b) What is the probability that a person chosen at random from those in this sample whose incomes are over $50,000 will be in the 31-45 age category? I found this to be .365, or 36.5%.
c) Bsed on your answers in parts a and b, is annual income independeent of age category for those in this sample? Explain. How do I prove this? I don't know what to do for this part.
It is not possible to critique answers a or b because the table is not here.
On c, again we need the table.
Here is the table:
Annual income
Age 25K-35K 35,001-50K Over 50K
21-30 8 15 27
31-45 22 32 35
46-60 12 14 27
Over 60 5 3 7
To answer part a of the question, you calculated the probability of selecting a person from the sample who falls into the 31-45 age category. You found that the probability is 0.43 or 43%. This means that if you randomly pick a person from the sample, there is a 43% chance they will be in the 31-45 age category.
To calculate this probability, you likely divided the number of people in the 31-45 age category by the total number of people in the sample. This is a correct approach. Let's assume you have a total of 100 people in the sample, and 43 of them belong to the 31-45 age category. In this case, the probability of selecting a person in the 31-45 age category would be 43/100 = 0.43.
Moving on to part b, you were asked to calculate the probability of selecting a person from the sample with an income over $50,000 who is also in the 31-45 age category. You found this probability to be 0.365 or 36.5%.
To compute this probability, you likely divided the number of people in both the 31-45 age category and with incomes over $50,000 by the total number of people with incomes over $50,000. Again, this is the correct approach. Let's assume you have a total of 50 people in the sample with incomes over $50,000, and 18 of them belong to the 31-45 age category. In this case, the probability of selecting a person who is both in the 31-45 age category and has an income over $50,000 would be 18/50 = 0.36 or 36%.
To address part c of the question, determining whether annual income is independent of age category for those in the sample requires analyzing the probabilities calculated in parts a and b. If the probability of being in the 31-45 age category (part a) is the same as the probability of being in the 31-45 age category given an income over $50,000 (part b), then we can conclude that annual income is independent of age category in this sample.
In your case, the probability of being in the 31-45 age category (part a) is 0.43 or 43%, while the probability of being in the 31-45 age category given an income over $50,000 (part b) is 0.365 or 36.5%. Since these probabilities are not the same, we can infer that annual income is not independent of age category in this sample.
To summarize, to prove independence between two variables, you compare the probabilities calculated for each variable independently and then conditional on the other variable. If the probabilities are not equal, it suggests a relationship between the variables, indicating that they are not independent.