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)
a) To find the probability that a person chosen at random from the sample is in the 31-45 age category, we need to divide the number of people in that age category by the total number of people in the sample.
Number of people in the 31-45 age category = 89
Total number of people in the sample = 207
Probability = Number of people in 31-45 age category / Total number of people in the sample
Probability = 89 / 207
b) To find the probability that a person chosen at random from the sample, whose income is over $50,000, is in the 31-45 age category, we need to consider only those people in the over $50,000 income category and find the ratio of people in the 31-45 age category.
Number of people in the 31-45 age category with income over $50,000 = 27
Number of people with income over $50,000 = 96
Probability = Number of people in 31-45 age category with income over $50,000 / Number of people with income over $50,000
Probability = 27 / 96
c) To determine if annual income is independent of age category, we need to compare the probabilities found in parts a) and b). If the probabilities are similar, then annual income is independent of age category. If the probabilities are different, then annual income is dependent on age category.
Compare the probabilities from parts a) and b). If they are similar, annual income is independent of age category. If they are different, annual income is dependent on age category.
Now let's move on to the next problem:
a) The probability that the student is female can be found by summing up the probabilities of being female for all age categories.
Probability of being female = Sum of probabilities of being female for all age categories
Probability of being female = 0.01 + 0.30 + 0.13 + 0.09
b) The probability that the student is age 25-34 can be found by summing up the probabilities of being in the 25-34 age category for both genders.
Probability of being age 25-34 = Probability of being age 25-34 for males + Probability of being age 25-34 for females
Probability of being age 25-34 = 0.12 + 0.13
c) The probability that the student is either male or at least 35 years old can be found by summing up the probabilities of being male and being at least 35 years old for both genders.
Probability of being male or at least 35 years old = Probability of being male + Probability of being at least 35 years old for males + Probability of being at least 35 years old for females
Probability of being male or at least 35 years old = 0.01 + 0.04 + 0.09
d) The probability that the student is at least 35 years old can be found by summing up the probabilities of being at least 35 years old for both genders.
Probability of being at least 35 years old = Probability of being at least 35 years old for males + Probability of being at least 35 years old for females
Probability of being at least 35 years old = 0.04 + 0.09
e) To determine if being a female and being at least 35 years old are independent events, we need to compare the probability of being female and at least 35 years old (calculated by multiplying the probability of being female and the probability of being at least 35 years old) with the probability of being female and at least 35 years old (calculated independently by adding the probabilities of being female and the probabilities of being at least 35 years old).
If the two probabilities are similar, then being a female and being at least 35 years old are independent events. If the two probabilities are different, then being a female and being at least 35 years old are dependent events.
A) 0.53/1.00=0.53
B) 0.25/1.00=0.25
C) 0.47+35=.82
D) 0.04 +0.35=.39
IS THIS CORRECT
Looks fine to me.
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
However, I will start you out.
10. Since you are dealing with percents, the grand total = 1.00, so you don't need to divide by the grand total as you would with raw scores.
a. female total
b. 25-34 total
c. Either-or probabilities are found by adding the individual probabilities.
Male total + (+35) total
d. +35 total
e. If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.