STAT200 Homework WEEK 5A++

STAT200 Homework WEEK 5
1. Muscular dystrophy is an incurable muscle-wasting disease. If a woman has a child with muscular dystrophy the probability is 2/3 that she is a carrier and 1/3 that she is not a carrier. There is a screening test called the CK test that is positive with probability 0.7 if a woman is a carrier and with probability 0.1 if she is not. Toni has a son with muscular dystrophy. She takes the CK test. Let’s use the following notation:
C: Toni is a carrier;
NC: Toni is not a carrier;
CKP: Toni gets a positive CK test result;
CKN: Toni gets a negative CK test result.
(a) Please determine the following probabilities: P(CKP|C), P(CKP|NC), P(C), P(NC).
(b) Calculate the marginal probability P(CKP)
(c) Calculate the conditional probability P(C|CKP)
2. *The scores of a state’s high school seniors on the ACT college entrance examination in a recent year had mean μ=22.3 and standard deviation σ=5.2. The distribution of scores is roughly Normal.
(a) What is the approximate probability that a random selected single student from those taking the test scores 26 or higher?
(b) Now consider a random sample of 20 students who took the test, what are the mean and standard deviation of the sample mean score x􀴤 of these 20 students.
(c) What is the approximate probability that the sample mean score x􀴤 of these 20 students is 26 or larger?
(d) *R bonus: use R to calculate (a)-(c). Provide your code and results.
3. * A $1 bet on a single number on a casino’s roulette wheel pays $35 if the ball ends up in the slot you choose. Here is the distribution of the payoff X:
Payoff X $0 $35
Probability 0.974 0.026
(a) What are the mean and standard deviation of X?
(b) Denny comes to the casino weekly during a long time and bets 400 spins in total. What’s the approximate average payoff for Denny? (Hint: Law of large numbers)
(c) What’s the standard deviation of the average payoff X
􀴥 for 400 spins?
(d) Calculate the approximate probability: P(X
􀴥>1), where X
􀴥 is the average payoff for 400 spins. (Hint: Central Limit Theorem).
(e) *R Bonus: Use R to simulate 100 spins and provide the average payoff.
4. In the test of ESP, the experimenter looks at cards that are hidden from the subject. Each card contains either a star, a circle or a square. As the experimenter looks at each 18 cards in turn, the subject tries to name the shape on the card.
(a) If a subject randomly guesses the shape on the card, what is the probability of a successful guess on a single card?
(b) Because the cards are independent, the count of successes in 18 cards has a binomial distribution. What is the probability that a subject correctly guesses the shape on at least 10 of the 18 cards, still assuming random guessing?
(c) If the experimenter looks at 90 cards instead of 18, what are the mean and standard deviation of the proportion of the successful guesses?
(d) With 90 card guesses, what is the approximate probability that subject has at least 36 correct random guesses?
5. The Census Bureau says that the 10 most common names in the United States are (in order) Smith, Johnson, Williams, Brown, Jones, Miller, Davis, Garcia, Rodriguez and Wilson. These names account for 4.9% of all U.S. residents. Out of curiosity, you look at the authors of the textbooks for your current courses. There are 12 authors in all. Would you be surprised if none of the names of these authors were among the 10 most common? Find the probability that none of the 12 textbook authors has one of the 10 most common names, assuming textbook author names are randomly selected from the population.