Show all your work. Indicate clearly the methods you use because you will be graded on the correctness of your methods as well as on the accuracy of your results and explanation.

A simple random sample of adults living in a suburb of a large city was selected. The age and annual income of each adult in the sample were recorded. The resulting data are summarized in the table below.

Annual Income
Age Category $25,000–$35,000 $35,001–$50,000 Over $50,000 Total
21–30 8 15 27 50
31–45 22 32 35 89
46–60 12 14 27 53
Over 60 5 3 7 15
Total 47 64 96 207

a) What is the probability that a person chosen at random from those in this sample will be in the 31–45 age category?
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? Show your work.
c) Based on your answers to (a) and (b), is annual income independent of age category for those in this sample? Explain. (7 points)

10. The following table is the assignment of probabilities that describes the age (in years) and the gender of a randomly selected American student.

Age 14–17 18–24 25–34 +35
Male 0.01 0.30 0.12 0.04
Female 0.01 0.30 0.13 0.09

a) What is the probability that the student is female?
b) What is the probability that the student is age 25–34?
c) What is the probability that the student is either male or at least 35 years old?
d) What is the probability that the student is at least 35 years old?
e) Are being a female and being at least 35 years old independent events? Provide statistical reasoning for your answer. (7 points)

Essay

2+2

To solve these problems, we will use the principles of probability and the given data from the tables.

Problem 1:
a) To find the probability that a person chosen at random from the sample will be in the 31-45 age category, we need to calculate the number of adults in that age category divided by the total number of adults in the sample.
Given that the number of adults in the 31-45 age category is 89 and the total number of adults is 207, the probability is 89/207.

b) To find 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, we need to calculate the number of adults in the intersection of both categories (31-45 age category and income over $50,000). This is given as 35 in the table.
Now, we find the probability by dividing the number of adults in the intersection by the total number of adults with income over $50,000.
Given that the number of adults with income over $50,000 is 96, the probability is 35/96.

c) To determine if annual income is independent of age category for those in this sample, we can compare the probabilities from parts (a) and (b). If the probabilities are approximately equal, then we can conclude that the two variables are independent.
In part (a), the probability of being in the 31-45 age category was 89/207, while in part (b), the probability of being in the 31-45 age category given income over $50,000 was 35/96. Since these probabilities are different, we can conclude that annual income is not independent of age category for those in this sample.

Problem 2:
a) The probability that the student is female is the sum of the probabilities in the "Female" row of the table.
P(Female) = 0.01 + 0.30 + 0.13 + 0.09 = 0.53

b) The probability that the student is age 25-34 is the sum of the probabilities in the "25-34" column of the table.
P(Age 25-34) = 0.12 + 0.13 = 0.25

c) To find the probability that the student is either male or at least 35 years old, we need to add the probabilities in the "Male" row and the "+35" column, then subtract the overlapping probability (male and at least 35 years old).
P(Male or at least 35 years old) = (0.01 + 0.30 + 0.12 + 0.04) + (0.04 + 0.09) - 0.04 = 0.56

d) The probability that the student is at least 35 years old is the sum of the probabilities in the "+35" column.
P(at least 35 years old) = 0.04 + 0.09 = 0.13

e) To determine if being a female and being at least 35 years old are independent events, we need to compare the probability of the intersection (female and at least 35 years old) with the product of the individual probabilities.
P(Female and at least 35 years old) = 0.04
P(Female) = 0.53
P(at least 35 years old) = 0.13

If the product of the individual probabilities (P(Female) * P(at least 35 years old)) is equal to the probability of the intersection (P(Female and at least 35 years old)), then the events are independent.
0.13 * 0.53 = 0.06 which is not equal to 0.04. Therefore, being a female and being at least 35 years old are dependent events.

Please note that the essay question requires a more detailed and expanded response, which might not be suitable to be answered in a short-format text like this.