ASSIGNMENT 2. Question 4.1 In each of the following situations, describe a sample space S for the random phenomenon.

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1 ASSIGNMENT 2 MGCR 271 SUMMER DUE THURSDAY, MAY 21, 2009 AT 18:05 IN CLASS Question 4.1 In each of the following situations, describe a sample space S for the random phenomenon. (1) A new business is started. After two years, it is either still in business or it had closed. (2) A rust prevention treatment is applied to a new car. The response variable is the length of time before rust begins to develop on the vehicle. (3) A student enrolls in a statistics course and at the end of a semester receives a letter grade. (4) A quality inspector examines four portable MP3 players and rates each as either acceptable or unacceptable. You record the sequence of ratings. (5) A quality inspector examines four portable MP3 players and rates each as either acceptable or unacceptable. You record the number of units rated as acceptable. (6) Choosing a random student in the class, the amount of time that student has spent studying in the past 24 hours. (7) The Physicians Health Study asked 11,000 physicians to take an aspirin every other day and observed how many of them had a hear attack in a 5-year period. (8) In a test of new package design, you drop a carton of a dozen eggs from a height of 1 foot and count the number of broken eggs. (9) A nutrition researcher feeds a new diet to a young male white rat. The response variable is the weight (in grams) that the rat gains in 8 weeks. Question 4.2 In a Feb 2007 buying-intention survey, the investment bank Goldman-Sachs found that 71% of respondents indicated interest in buying an Apple mobile phone. Of the respondents in interested in buying an Apple mobile phone, 15% indicated that they would switch cellular carriers to get an Apple mobile phone. Consider randomly selecting one survey respondent. Let A be the event that the respondent is interested in buying an Apple mobile phone, 1

2 2 MGCR 271 SUMMER DUE THURSDAY, MAY 21, 2009 AT 18:05 IN CLASS and let B be the event that the respondent would switch carriers to get an Apple mobile phone. (1) Draw a Venn diagram which shows the relationship between the events A and B. (2) What is the probability that that a respondent is interested in buying an Apple mobile phone or would be willing to switch carriers to get an Apple mobile phone. (3) Calculate the probability that a respondent is interested in buying an Apple mobile phone and would be willing to switch carriers to get an Apple mobile phone. Are A and B independent? Question 4.3 (not to hand in) Ramon has applied to MBA programs at both Harvard and Stanford. He thinks the probability that Harvard will admit him is 0.4, the probability that Stanford will admit him is 0.5, and the probability that both will admit him is 0.2. (1) Make a Venn diagram with the probabilities given marked on the Venn diagram. (2) What is the probability that Ramon get rejected from both universities? (3) What is the probability that he gets into Stanford but not Harvard? Question 4.4 Functional Robotics Corporation buys electrical controllers from a Japanese supplier. The company s treasurer thinks that there is probability of 0.4 that the dollar will fall in value against the Japanese Yen in the next month. The treasurer also believes that if the dollar falls there is a probability of 0.8 that the supplier will demand renegotiation of the contract. What probability has the treasurer assigned to the event that the dollar falls and the supplier demands renegotiation? Question 4.5 Consolidated Builders has bid on two large construction projects. The company president believes that the probability of winning the first contract (event A) if 0.6, that the probability of winning the second contract (event B) is 0.5, and the probability of winning both contracts is 0.3. (1) What is the probability that Consolidated Builders will win at least one contract? (2) Are A and B independent events? Show your calculation which proves dependence or independence. (3) Draw a Venn diagram that illustrates the relationship between A and B. (4) Compute the probability that Consolidated Builders wins both jobs. (5) Compute the probability that Consolidated Builders wins the first job but not the second.

3 ASSIGNMENT 2 3 (6) Compute the probability that Consolidated Builders wins the second job but not the first. (7) Compute the probability that Consolidated Builders does not win either job?. Question 4.6 Government data show that 27% of employed people have at least 4 years of college and that 14% of employed people work as laborers or operators of machines or vehicles. Can you conclude - because (0.27)(0.14) = that about 3.8% of employed people are college-educated laborers or operators? Explain your answer. Question 4.7 (not to hand in) An employee suspected of having used an illegal drug is given two tests that operate independently of each other. Test A has a probability of 0.9 of being positive if the illegal drug has been used. Test B has the probability of 0.8 of being positive if the illegal drug as been used. What is the probability that neither test is positive if the illegal drug has been used? Question 4.8 Employees sometimes install on their home computers software that was purchased by their employer for use on their work computers. For most commercial software packages, this is illegal. Suppose that 5% of all employees at a large corporation have illegally installed corporate software on their home computers knowing the act is illegal and an additional 2% have installed corporate software on their home computers not knowing it was illegal. Of the 5% aware that the home installation is illegal, 80%, will deny that they knew the act was illegal if confronted by a software auditor. If an employee who has illegally installed software at home is confronted and denies knowing it was an illegal act, what is the probability that the employee knew the home installation was illegal? Question 4.9 The accuracy of a medical diagnostic test, in which a positive result indicates the presence of a disease, is often states in terms of its sensitivity, the proportion of diseased people that test positive or P (+ Disease), and it specificity, the proportion of diseased people without the disease who test negative or P ( No Disease). Suppose that 10% of the population has the disease (called the prevalence rate. A diagnostic test for the disease has 99% sensitivity and 98% specificity, therefore, P (+ Disease) = 0.99, P ( No Disease) = 0.98, P ( Disease) = 0.01 and P (+ No Disease) = 0.02 (1) A person s test result is positive. What is the probability that the person actually has the disease? (2) A person s test result is negative. What is the probability that the person actually does not have the disease? Considering this results and the previous result, would you say that the diagnostic test is reliable? Why or why not?

4 4 MGCR 271 SUMMER DUE THURSDAY, MAY 21, 2009 AT 18:05 IN CLASS (3) Now suppose that the disease is rare with a prevalence rate of 0.1%. Using the same diagnostic test, what is the probability that the person who test positive actually has the disease? (4) The results from (1) and (3) are based on the same diagnostic test applied to populations with very different prevalence rates. Does this suggest any reason why mass screening programs should not be recommended for a rare disease? Explain.

5 Question 5.1 ASSIGNMENT 2 5 A study of class mobility in England looked at the economic class reached by the sons of lower-class fathers. Economic classes are numbered from 1 (low) to 5 (high). Take X to be the class of a randomly chosen son of a father in Class 1. The study found that the probability model for X is Son s Class X Probability (1) Verify that the distribution satisfies the two requirements for a legitimate assignment of probabilities to individual outcomes (2) Compute P (X 3) (3) Computer P (X < 3) (4) Write the event a son of a lower-class father reaches one of the two highest classes in terms of value of X. What is the probability of this event? Question 5.2 Each week your business receives orders from customers who wish to have air-conditioning units installed in their homes. From past data, you estimate the distribution of the number of units ordered in a week X to be Units Ordered X Probability (1) Sketch a probability histogram for the distribution of X. (2) Compute the mean of X and mark it on the horizontal axis of your probability histogram. Compute the variance. (3) If you hire enough workers to be able to handle the mean demand, what is the probability that you will be unable to handle all of a week s installation orders? Question 5.3 The distribution of the count X of tossing heads in four tosses of a balanced coin was found to be Number of Heads x i Probability p i Find the mean µ X from this distribution. Find the mean number of heads for a single coin toss and show that your two results are related by the addition property of the mean ( µ X+Y = µ X + µ Y ).

6 6 MGCR 271 SUMMER DUE THURSDAY, MAY 21, 2009 AT 18:05 IN CLASS Question 5.4 You have two scales for measuring weight. Both scales give answers that vary a bit in repeated weighings of the same item. If the true weight of an item is 2 grams (g), the first scale produces readings X that gave a mean of 2.000g and a standard deviation of 0.002g. The second scale s readings Y have mean 2.001g and standard deviation 0.001g. (1) What are the mean and standard deviation of the difference Y X between the reading (the readings Y and X are independent)? (2) You measure once with each scale and average the readings. Your results is Z = (X + Y/2). What is µ Z and σ Z? Is the average Z more or less variable that the reading Y of the less variable scale? Question 5.5 (not to hand in) You purchase a hot stock for $1000. The stock either gains 30% or loses 25% each day, each with equal probability. The stock s returns on consecutive days are independent of each other. This implies that all four possible combinations of gains and loses in two days are equally likely, each having probability You plan to sell the stock after two days. (1) What are the possible values of the stock after two days, and what is the probability for each value? What is the probability that the stock is worth more after two days that the $1000 you paid for it? (2) What is the mean value of the stick after two days? Comment on how the interpretation of the mean differs from the interpretation of the response obtained in the previous question. Question 5.6 You Observe the gender of the next 20 children born at a local hospital; X is the number of girls among them. Does X have a binomial distribution? Provide your rationale. Question 5.7 A factory employs several thousand workers, of whom 30% are Hispanic. executive committee members were chosen from the workers at random. A 15 union (1) What is the probability that exactly 3 members of the committee are Hispanic? (2) What is the probability that 3 or fewer members of the committee are Hispanic? (3) What is the mean number of Hispanics on randomly chosen committees of 15 workers? (4) What is the standard deviation σ of the count X of Hispanic members? (5) Suppose that 10% of the factory workers were Hispanic. What is σ in this case? What is σ if 1% of factory workers are Hispanic? What does your work show about

7 ASSIGNMENT 2 7 the behavior of the standard deviation of a binomial distribution as the probability of a success gets closer to 0? Question 5.8 Let Y be a uniformly-distributed random number between 0 and 1. Find the following probabilities. (1) P (0 Y 0.4) (2) P (0.4 Y 1.0) (3) P (0.3 Y 0.5) (4) P (0.3 < Y < 0.5) (5) P (0.226 Y 0.713) Question 5.9 Some health insurance companies treat pregnancy as a preexisting condition when it comes to paying for maternity expenses for a new policyholder. Sometimes the exact date of conception is unknown so the insurance company must count back from the expected due date to judge whether or not conception occurred before or after the new policy began. The length of human pregnancies from conception to birth varies according to a Normal distribution with mean 266 days and and a standard deviation of 16 days. (1) Between what values of the length of the middle 95% of all pregnancies fall? (2) How short are the shortest 2.5% of all pregnancies? (3) How likely is it that a woman with an expected due date 218 days after her policy began conceived the child after her policy began? (4) What percentage of pregnancies last more than 240 days? (5) What percent of pregnancies last between 240 and and 270 days? (6) How long do the longest 20% of pregnancies last? Question 5.10 (1) Find the number z such that the proportion of observations that are less than z in a standard Normal statistical distribution is 0.8. (2) Find the bumber z such that 35% of all observations from a standard Normal distribution are greater than z. Question 5.11 (1) A gambler knows that red and black are equally likely to occur on each spin of a roulette wheel. He observes five consecutive reds and bets heavily on red at the

8 8 MGCR 271 SUMMER DUE THURSDAY, MAY 21, 2009 AT 18:05 IN CLASS next spin. Asked why, he says that red is hit and that the run of reds is likely to continue. Explain to the gambler what is wrong with this reasoning. (2) After hearing your explain explanation, the gambler moves to a poker table. He is dealt five straight red cards from a single 52-card deck. He assumes that the next card dealt to him is equally likely to be red or black. Is the gambler correct? Why or why not? Question 5.12 Sodium content (in milligrams) in measured for bags of potato chips sampled from a production line. The standard deviation of the sodium content measurements is σ = 10 mg. The sodium content is measure 3 times and the mean x of the 3 measurements is recorded. (1) What is the standard deviation of the mean result? (that is to say if you kept making 3 measurements and averaging them, what would be the standard deviation of all x s?) (2) How many times must we repeat the measurement to reduce the standard deviation of x to 5? Explain why there is an advantage of reporting the average of several measurements rather than the result of a single measurement. Question 5.13 The scores of students on the ACT college entrance examination in a recent year had the Normal distribution with mean µ = 18.6 and standard deviation σ = 5.9. (1) What is the probability that a single student randomly chosen from all those taking the test scores 21 or higher? (2) Now take an SRS of 50 students who took the test. What are the mean and standard deviation of the sample mean score x? (3) What is the probability that the mean score x of these students is 21 or higher?