Stat 2000, Section 001, Homework Assignment 12 (Due 12/2/2002 11:59pm)

• 0) Reading: Section 5.1, 5.2
  
  
• 1) Please work on the following textbook exercises from Moore/McCabe (3rd Edition):
  
  
  • Exercise 5.1 (1 Point):
    For each of the following situations, indicate whether a binomial distribution is a reasonable probability model for the random variable X. Give your reasons in each case.
    (a) You observe the sex of the next 50 children born at a local hospital; X is the number of girls among them.
    (b) A couple decides to continue to have children until their first girl is born; X is the total number of children the couple has.
    (c) You want to know what percent of married people believe that mothers of young children should not be employed outside the home. You plan to interview 50 people, and for the sake of convenience you decide to interview both the husband and the wife in 25 married couples. The random variable X is the number among the 50 persons interviewed who think mothers should not be employed.
    
    
  • Exercise 5.2 (1 Point):
    In each situation below, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in each case.
    (a) An auto manufacturer chooses one car from each hour's production for a detailed quality inspection. One variable recorded is the count X of finish defects (dimples, ripples, etc.) in the car's paint.
    (b) The pool of potential jurors for a murder case contains 100 persons chosen at random from the adult residents of a large city. Each person in the pool is asked whether he or she opposes the death penalty; X is the number who say ``Yes.''
    (c) Joe buys a ticket in his state's ``Pick 3'' lottery game every week; X is the number of times in a year that he wins a prize.
    
    
  • Exercise 5.4 (1 Point):
    Some of the methods in this section are approximations rather than exact probability results. We have given rules of thumb for safe use of these approximations.
    (a) You are interested in attitudes toward drinking among the 75 members of a fraternity. You choose 25 members at random to interview. One question is ``Have you had five or more drinks at one time during the last week?'' Suppose that in fact 20% of the 75 members would say ``Yes.'' Explain why you cannot safely use the Bin(25, 0.2) distribution for the count X in your sample who say ``Yes.''
    (b) The National AIDS Behavioral Surveys found that 0.2% (that's 0.002 as a decimal fraction) of adult heterosexuals had both received a blood transfusion and had a sexual partner from a group at high risk of AIDS. Suppose that this national proportion holds for your region. Explain why you cannot safely use the normal approximation for the sample proportion who fall in this group when you interview an SRS of 500 adults.
    
    
  • Exercise 5.6 (1 Point):
    You operate a restaurant. You read that a sample survey by the National Restaurant Association shows that 40% of adults are committed to eating nutritious food when eating away from home. To help plan your menu, you decide to conduct a sample survey in your own area. You will use random digit dialing to contact an SRS of 200 households by telephone.
    (a) If the national result holds in your area, it is reasonable to use the Bin(200, 0.4) distribution to describe the count X of respondents who seek nutritious food when eating out. Explain why.
    (b) What is the mean number of nutrition-conscious people in your sample if p = 0.4 is true? What is the probability that X lies between 75 and 85? (Use software or the normal approximation.)
    (c) You find 100 of your 200 respondents concerned about nutrition. Is this reason to believe that the percent in your area is higher than the national 40%? To answer this question, find the probability that X is 100 or larger if p = 0.4 is true. If this probability is very small, that is reason to think that p is actually greater than 0.4.
    
    
  • Exercise 5.11 (1 Point):
    According to government data, 25% of employed women have never been married.
    (a) If 10 employed women are selected at random, what is the probability that exactly 2 have never been married? (Use the binomial probability formula.)
    (b) What is the probability that 2 or fewer have never been married?
    (c) What is the probability that at least 8 have been married?
    
    
  • Exercise 5.12 (1 Point):
    A university that is better known for its basketball program than for its academic strength claims that 80% of its basketball players get degrees. An investigation examines the fate of all 20 players who entered the program over a period of several years that ended 5 years ago. Of these players, 11 graduated and the remaining 9 are no longer in school. If the university's claim is true, the number of players who graduate among the 20 studied should have the Bin(20, 0.8) distribution.
    (a) Find the probability that exactly 11 players graduate under these assumptions. (Use the binomial probability formula.)
    (b) Find the probability that 11 or fewer players graduate. This probability is so small that it casts doubt on the university's claim.
    
    
  • Exercise 5.18 (2 Points):
    According to government data, 21% of American children under the age of six live in households with incomes less than the official poverty level. A study of learning in early childhood chooses an SRS of 300 children.
    (a) What is the mean number of children in the sample who come from poverty-level households? What is the standard deviation of this number?
    (b) Use the normal approximation to calculate the probability that at least 80 of the children in the sample live in poverty. Be sure to check that you can safely use the approximation.
    
    
  • Exercise 5.20 (2 Points):
    A selective college would like to have an entering class of 1200 students. Because not all students who are offered admission accept, the college admits more than 1200 students. Past experience shows that about 70% of the students admitted will accept. The college decides to admit 1500 students. Assuming that students make their decisions independently, the number who accept has the Bin(1500, 0.7) distribution. If this number is less than 1200, 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 1000 students accept.
    (c) The college does not want more than 1200 students. What is the probability that more than 1200 will accept?
    (d) If the college decides to increase the number of admission offers to 1700, what is the probability that more than 1200 will accept?
    
    
  • Exercise 5.21 (2 Points):
    Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) The correctness of answers to different questions are independent. Jodi is a good student for whom p = 0.75.
    (a) Use the normal approximation to find the probability that Jodi scores 70% or lower on a 100-question test.
    (b) If the test contains 250 questions, what is the probability that Jodi will score 70% or lower?
    (c) How many questions must the test contain in order to reduce the standard deviation of Jodi's proportion of correct answers to half its value for a 100-item test?
    (d) Laura is a weaker student for whom p = 0.6. Does the answer you gave in (c) for the standard deviation of Jodi's score apply to Laura's standard deviation also?
    
    
  • Exercise 5.24 (1 Point):
    Juan makes a measurement in a chemistry laboratory and records the result in his lab report. The standard deviation of students' lab measurements is sigma = 10 milligrams. Juan repeats the measurement 3 times and records the mean xbar of his 3 measurements.
    (a) What is the standard deviation sigma_xbar of Juan's mean result?
    (b) How many times must Juan repeat the measurement to reduce the standard deviation of xbar to 5? Explain to someone who knows no statistics the advantage of reporting the average of several measurements rather than the result of a single measurement.
    
    
  • Exercise 5.26 (1 Point):
    The scores of students on the ACT college entrance examination in a recent year had the normal distribution with mean mu = 18.6 and standard deviation sigma = 5.9.
    (a) What is the probability that a single student randomly chosen from all those taking the test scores 21 or higher?
    (b) Now take an SRS of 50 students who took the test. What are the mean and standard deviation of the sample mean score xbar of these 50 students?
    (c) What is the probability that the mean score xbar of these students is 21 or higher?
    
    
  • Exercise 5.30 (1 Point):
    Judy's doctor is concerned that she may suffer from hypokalemia (low potassium in the blood). There is variation both in the actual potassium level and in the blood test that measures the level. Judy's measured potassium level varies according to the normal distribution with mu = 3.8 and sigma = 0.2. A patient is classified as hypokalemic if the potassium level is below 3.5.
    (a) If a single potassium measurement is made, what is the probability that Judy is diagnosed as hypokalemic?
    (b) If measurements are made instead on 4 separate days and the mean result is compared with the criterion 3.5, what is the probability that Judy is diagnosed as hypokalemic?
    
    
  • Exercise 5.34 (1 Point):
    The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.4. This distribution takes only whole-number values, so it is certainly not normal.
    (a) Let xbar be the mean number of accidents per week at the intersection during a year (52 weeks). What is the approximate distribution of xbar according to the central limit theorem?
    (b) What is the approximate probability that xbar is less than 2?
    (c) What is the approximate probability that there are fewer than 100 accidents at the intersection in a year? (Hint: Restate this event in terms of xbar.)
    
    
• 2) Recall the behavior of the manipulated coin (70% chance of head) we discussed in class. Suppose we toss this coin 1000 times. (3 Points)
  
  
  • a) What are mean, variance, and standard deviation for a random variable X that is related to this coin?
    
    
  • b) What is the probability to end up with 700 to 900 times head with this coin? Use an appropriate approximation to calculate this probability.
    
    
  • c) And what is the probability to end up with 700 to 900 times head with a "fair" coin? Use an appropriate approximation again to calculate this probability.
    
    
• 3) Look at the Central Limit Theorem applet again at
  http://www.stat.sc.edu/~west/javahtml/CLT.html
  
  Roll 3 dice once, 10 times, 50 times, 100 times, 500 times and decribe your observations. How does this compare to rolling 1 die the same number of times? And what happens if you roll 5 dice the same number of times? (3 Points)
  
  
• 4) Work with the Normal Approximation to Binomial applet at
  http://www.ruf.rice.edu/~lane/stat_sim/normal_approx/index.html
  
  Select N = 5 and set "from" = 2 and "to" = 4. Then use p = 0.1, 0.5, and 0.8, and report what the exact binomial probabilities and the probabilities by the normal approximation are.
  
  Similarly, work with N = 50, "from" = 20 and "to" = 30. Use p = 0.4, 0.5, and 0.7 here. Report the probabilities.
  
  Finally, look at N = 500, "from" = 240, "to" = 260, and p = 0.5 and report the probabilities.
  
  Experiment with some other parameters and summarize your findings in two or three sentences. (3 Points)
  
  

Happy Thanksgiving !!!