Part A - Questions -
Solve the all questions. All work must be shown.
Question 1 - Distribution of blood types - All human blood can be "ABO-typed" as one of O, A, B, or AB, but the distribution of the types varies a bit among groups of people. Here is the distribution of blood types for a randomly chosen person in the United States:
|
Blood type
|
A
|
B
|
AB
|
O
|
|
U.S. probability
|
0.42
|
0.11
|
?
|
0.44
|
(a) What is the probability of type AB blood in the United States?
(b) Maria has type B blood. She can safely receive blood transfusions from people with blood types O and B. What is the probability that a randomly chosen person from the United States can donate blood to Maria?
Question 2 - French and English in Canada.
Canada has two official languages, English and French. Choose a Canadian at random and ask, "What is your mother tongue?" Here is the distribution of responses, combining many separate languages from the broad Asian/Pacific region:
|
Language
|
English
|
French
|
Asian/Pacific
|
Other
|
|
Probability
|
0.59
|
?
|
0.07
|
0.11
|
(a) What probability should replace "?" in the distribution?
(b) What is the probability that a Canadian's mother tongue is not English? Explain how you computed your answer.
Question 3 - Texas hold 'em - The game of Texas hold 'em starts with each player receiving two cards. Here is the probability distribution for the number of aces in two-card hands:
|
Number of aces
|
0
|
1
|
2
|
|
Probability
|
0.8507
|
0.1448
|
0.0045
|
(a) Verify that this assignment of probabilities satisfies the requirement that the sum of the probabilities for a discrete distribution must be 1.
(b) Make a probability histogram for this distribution.
(c) What is the probability that a hand contains at least one ace? Show two different ways to calculate this probability.
Question 4 - Normal approximation for a sample proportion.
A sample survey contacted an SRS of 700 registered voters in Oregon shortly after an election and asked respondents whether they had voted. Voter records show that 56% of registered voters had actually voted. We will see in the next chapter that in this situation the proportion p^ of the sample who voted has approximately the Normal distribution with mean μ = 0.56 and standard deviation σ = 0.019.
(a) If the respondents answer truthfully, what is P(0.52 ≤ p^ ≤ 0.60)? This is the probability that the statistic p^ estimates the parameter 0.56 within plus or minus 0.04.
(b) In fact, 72% of the respondents said they had voted (p^ = 0.72). If respondents answer truthfully, what is P(p^ ≥ 0.72)? This probability is so small that it is good evidence that some people who did not vote claimed that they did vote.
Question 5 - Mean of the distribution for the number of aces.
Examined the probability distribution for the number of aces when you are dealt cards in the game of Texas hold 'em. Let X represents number of aces in a randomly selected deal of two in this game. Here is the probability distribution the random variable X:
|
Value of X
|
0
|
1
|
2
|
|
Probability
|
0.8507
|
0.1148
|
0.0045
|
Find μX, the mean of the probability distribution of X.
Question 6 -
Standard deviation of the number of aces. Refer Question 5. Find the standard deviation of the number of aces.
Question 7 - Education and income. Call a household prosperous if its income exceeds $100,000. Call the household educated if the householder completed college. Select an American household at random, and let A be the event that the selected household is prosperous and B the event that it is educated. According to the Current Population Survey, P(A) = 0.138, P(B) = 0.261, and the probability that a household is both prosperous and educated is P(A and B) = 0.082. What is the probability P(A or B) that the household selected is either prosperous or educated?
Question 8 - Academic degrees and gender. Here are the projected numbers (in thousands) of earned degrees in the United States in the 2010-2011 academic year, classified by level and by the sex of the degree recipient?
|
|
Bachelor's
|
Master's
|
Professional
|
Doctorate
|
|
Female
|
933
|
502
|
51
|
26
|
|
Male
|
661
|
260
|
44
|
26
|
(a) Convert this table to a table giving the probabilities for selecting a degree earned and classifying the recipient by gender and the degree by the levels given above.
(b) If you choose a degree recipient at random, what is the probability that the person you choose is a woman?
(c) What is the conditional probability that you choose woman, given that the person chosen received a professional degree?
(d) Are the events "choose a woman" and "choose a professional degree recipient's independent? How do you know?
Question 9 - Grades in a math course. Indiana University posts the grade distributions for its courses online. Students in one section of Math 118 in the fall 2012 semester received 33% A's, 33% B's, 20% C's, 12% D's, and 2% F's.
(a) Using the common scale A = 4, B = 3, C = 2, D = 1, F = 0, take X to be the grade of a randomly chosen 118 student. Use the definitions of the mean and standard deviation for discrete random variables to find the mean p. and the standard deviation of grades in this course.
(b) Math 118 is a large enough course that we can take the grades of an SRS of 25 students to be independent of each other. If x- is the average of these 25 grades, what are the mean and standard deviation of x-?
(c) What is the probability that a randomly chosen Math 118 student gets a B or better, P(X ≥ 3)?
(d) What is the approximate probability P(x- ≥ 3) that the grade point average for 25 randomly chosen Math 118 students is B or better?
Question 10 - Paying for music downloads. A survey of Canadian teens aged 12 to 17 years reported that roughly 75% of them used a fee-based website to download music. You decide to interview a random sample of 15 U.S. teenagers. For now, assume that they behave similarly to the Canadian teenagers.
(a) What is the distribution of the number X who used afee-based website to download music? Explain your answer?
(b) What is the probability that at least 12 of the 15 teenagers your sample used a fee-based website to download music.
Question 11 - Paying for music downloads, continued. Refer to Question 10. Suppose that only 60% of the U.S. teenagers used a fee-based website to download music.
(a) If you interview 15 U.S. teenagers at random, what is the mean of the count X who used a fee-based website to download music? What is the mean of the proportion p^ in your sample who used a fee-based website to download music?
(b) Repeat the calculations in part (a) for samples of size 150 and 1500. What happens to the mean count of successes as the sample size increases? What happens to the mean proportion of successes?
Question 12 - Admitting students to college. A selective college would like to have an entering class of 950 students. Because not all students who are offered admission accept, the college admits more than 950 students. Past experience shows that about 75% of the students admitted will accept. The college decides to admit 1200 students. Assuming that students make their decisions independently, the number who accept has the B(1200, 0.75) distribution. If this number is less than 950, the college will admit students from its waiting list.
(a) What are the mean and the standard deviation of the number X of students who accept?
(b) Use the Normal approximation to find the probability that at least 800 students accept.
(c) The college does not want more than 950 students. What is the probability that more than 950 will accept?
(d) If the college decides to increase the number of admission offers to 1300, what is the probability that more than 950 will accept?
Part B - Stata Application
Suppose the probability of reporting fair or poor health (versus average, good, or excellent health) in the adult population in Missouri is .302. If you take a sample of 30, what is the likelihood that 5 or less will be obese? What is the likelihood that you take a sample of 30, what is the likelihood that 5 or less will be obese? What is the likelihood that 10 or less will be obese? Include your stat output and syntax with your interpretation.