Exercise Questions: Chapter What is wrong? Explain what is wrong in each of the following scenarios.

Transcription

1 5.9 What is wrong? Explain what is wrong in each of the following scenarios. (a) If you toss a fair coin three times and a head appears each time, then the next toss is more likely to be a tail than a head. (b) If you toss a fair coin three times and a head appears each time, then the next toss is more likely to be a head than a tail. (c) is one of the parameters for a binomial distribution What is wrong? Explain what is wrong in each of the following scenarios. (a) In the binomial setting X is a proportion. (b) The variance for a binomial count is. (c) The Normal approximation to the binomial distribution is always accurate when n is greater than Should you use the binomial distribution? 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. If a binomial distribution applies, give the values of n and p. (a) A poll of 200 college students asks whether or not you are usually irritable in the morning. X is the number who reply that they are usually irritable in the morning. (b) You toss a fair coin until a head appears. X is the count of the number of tosses that you make. (c) Most calls made at random by sample surveys don t succeed in talking with a live person. Of calls to New York City, only 1/12 succeed. A survey calls 500 randomly selected numbers in New York City. X is the number that reach a live person Typographic errors. Typographic errors in a text are either nonword errors (as when the is typed as teh ) or word errors that result in a real but incorrect word. Spellchecking software will catch nonword errors but not word errors. Human proofreaders catch 70% of word errors. You ask a fellow student to proofread an essay in which you have deliberately made 10 word errors. (a) If the student matches the usual 70% rate, what is the distribution of the number of errors caught? What is the distribution of the number of errors missed? (b) Missing 4 or more out of 10 errors seems a poor performance. What is the probability that a proofreader who catches 70% of word errors misses 4 or more out of 10? 5.15 Typographic errors. Return to the proofreading setting of Exercise 5.13.

2 (a) What is the mean number of errors caught? What is the mean number of errors missed? You see that these two means must add to 10, the total number of errors. (b) What is the standard deviation σ of the number of errors caught? (c) Suppose that a proofreader catches 90% of word errors, so that p = 0.9. What is σ in this case? What is σ if p = 0.99? What happens to the standard deviation of a binomial distribution as the probability of a success gets close to 1? 5.17 Typographic errors. In the proofreading setting of Exercise 5.13, what is the smallest number of misses m with P(X m) no larger than 0.05? You might consider m or more misses as evidence that a proofreader actually catches fewer than 70% of word errors Attitudes toward drinking and behavior studies. 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 30 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 30% of the 75 members would say Yes. Explain why you cannot safely use the B(30, 0.3) distribution for the count X in your sample who say Yes. (b) The National AIDS Behavioral Surveys found that 0.2% (that s 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 1000 adults A college alcohol study. The Harvard College Alcohol Study finds that 67% of college students support efforts to crack down on underage drinking. The study took a sample of almost 15,000 students, so the population proportion who support a crackdown is very close to p = The administration of your college surveys an SRS of 200 students and finds that 140 support a crackdown on underage drinking. (a) What is the sample proportion who support a crackdown on underage drinking? (b) If in fact the proportion of all students on your campus who support a crackdown is the same as the national 67%, what is the probability that the proportion in an SRS of 200 students is as large or larger than the result of the administration s sample? (c) A writer in the student paper says that support for a crackdown is higher on your campus than nationally. Write a short letter to the editor explaining why the survey does not support this conclusion.

3 5.33 Multiple-choice tests. 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 an answer to a question is independent of the correctness of answers to other questions. Jodi is a good student for whom p = (a) Use the Normal approximation to find the probability that Jodi scores 80% or lower on a 100-question test. (b) If the test contains 250 questions, what is the probability that Jodi will score 80% 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 = Does the answer you gave in (c) for the standard deviation of Jodi s score apply to Laura s standard deviation also? 5.41 What is wrong? Explain what is wrong in each of the following scenarios. (a) If the standard deviation of a population is 10, then the variance of the mean for an SRS of 20 observations from this population will be 10/. (b) When taking SRS s from a population, larger sample sizes will have larger standard deviations. (c) The mean of a sampling distribution changes when the sample size changes Cholesterol levels of teenagers. A study of the health of teenagers plans to measure the blood cholesterol level of an SRS of 13- to 16-year olds. The researchers will report the mean from their sample as an estimate of the mean cholesterol level µ in this population. (a) Explain to someone who knows no statistics what it means to say that unbiased estimator of µ. is an (b) The sample result is an unbiased estimator of the population truth µ no matter what size SRS the study chooses. Explain to someone who knows no statistics why a large sample gives more trustworthy results than a small sample Diabetes during pregnancy. Sheila s doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Sheila s measured glucose

4 level one hour after ingesting the sugary drink varies according to the Normal distribution with µ = 125 mg/dl and σ = 10 mg/dl. (a) If a single glucose measurement is made, what is the probability that Sheila is diagnosed as having gestational diabetes? (b) If measurements are made instead on 3 separate days and the mean result is compared with the criterion 140 mg/dl, what is the probability that Sheila is diagnosed as having gestational diabetes? 5.52 A lottery payoff. A $1 bet in a state lottery s Pick 3 game pays $500 if the threedigit number you choose exactly matches the winning number, which is drawn at random. Here is the distribution of the payoff X: Each day s drawing is independent of other drawings. (a) What are the mean and standard deviation of X? (b) Joe buys a Pick 3 ticket twice a week. What does the law of large numbers say about the average payoff Joe receives from his bets? (c) What does the central limit theorem say about the distribution of Joe s average payoff after 104 bets in a year? (d) Joe comes out ahead for the year if his average payoff is greater than $1 (the amount he spent each day on a ticket). What is the probability that Joe ends the year ahead? 5.53 Defining a high glucose reading. In Exercise 5.51, Sheila s measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with µ = 125 mg/dl and σ = 10 mg/dl. What is the level L such that there is probability only 0.05 that the mean glucose level of 3 test results falls above L for Sheila s glucose level distribution? 5.54 Flaws in carpets. The number of flaws per square yard in a type of carpet material varies with mean 1.5 flaws per square yard and standard deviation 1.3 flaws per square yard. This population distribution cannot be Normal, because a count takes only wholenumber values. An inspector studies 200 square yards of the material, records the number of flaws found in each square yard, and calculates, the mean number of flaws per square yard inspected. Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds 2 per square yard.

5 5.62 Investments in two funds. Linda invests her money in a portfolio that consists of 70% Fidelity 500 Index Fund and 30% Fidelity Diversified International Fund. Suppose that in the long run the annual real return X on the 500 Index Fund has mean 9% and standard deviation 19%, the annual real return Y on the Diversified International Fund has mean 11% and standard deviation 17%, and the correlation between X and Y is 0.6. (a) The return on Linda s portfolio is R = 0.7X + 0.3Y. What are the mean and standard deviation of R? (b) The distribution of returns is typically roughly symmetric but with more extreme high and low observations than a Normal distribution. The average return over a number of years, however, is close to Normal. If Linda holds her portfolio for 20 years, what is the approximate probability that her average return is less than 5%? (c) The calculation you just made is not overly helpful, because Linda isn t really concerned about the mean return. To see why, suppose that her portfolio returns 12% this year and 6% next year. The mean return for the two years is 9%. If Linda starts with $1000, how much does she have at the end of the first year? At the end of the second year? How does this amount compare with what she would have if both years had the mean return, 9%? Over 20 years, there may be a large difference between the ordinary mean and the geometric mean, which reflects the fact that returns in successive years multiply rather than add.