Statistics Probability Inference And Decision 1st Edition Robert L. Winkler, William L. Hays - Solutions

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4.10 SOC On the scale mentioned in problem 4.7, if a score of 40 or more is considered “highly discriminatory”what is the probability that a human resources manager selected at random will have a score in that range?
4.8 The average burglary rate for a jurisdiction has been 311 per year with a standard deviation of 50. What is the probability that next year the number of burglaries will bea. less than 250? _____b. less than 300? _____c. more than 350? _____d. more than 400? _____e. between 250 and 350? _____f.
4.7 A scale measuring ageism (age discrimination) has been administered to a large sample of human resources managers at major corporations. The distribution of scores is approximately normal with a mean of 31 and a standard deviation of 5. What percentage of the sample had scoresa. below 20?
4.5 At Algebra University the 200 students enrolled in Introductory Biology took a final exam on which their mean score was 72 and their standard deviation was 6. The table below presents the grades of 10 students. Convert each into a Z score and determine the number of people who scored higher or
4.3 A class of final-year students at a university has been given a comprehensive examination to assess their educational experience. The mean on the test was 74 and the standard deviation was 10. What percentage of the students had scoresa. between 75 and 85? _____b. between 80 and 85? _____c.
4.2 Assume that the distribution of a graduate-school entrance exam is normal with a mean of 500 and a standard deviation of 100. For each score below, find the equivalent Z score, the percentage of the area above the score, and the percentage of the area below the score.Xi Z Score % Area Above%
4.1 Scores on a quiz were normally distributed and had a mean of 10 and a standard deviation of 3. For each score below, find the Z score and the percentage of area above and below the score.Xi Z Score % Area Above % Area Below 56 78 911 12 14 15 16 18
3. Express areas under the curve in terms of probabilities.
2. Convert raw (original) scores to Z scores and use Z scores and the normal curve table (Appendix A) to find areas above, below, and between points on the curve.
1. Define and explain the concept of the normal curve.
3.15 At Algebra University the math department ran some special sections of the introductory math course using a variety of innovative teaching techniques. Students were randomly assigned to either the traditional sections or the experimental sections, and all students were given the same final
3.14 PA The data below represent the percentage of workers living in each city who used public transportation to commute to work in 2010 (fictitious data).City Percent Abbotsford 20 Barrie 32 Calgary 26 Dawson 6 Fredericton 9 Edmonton 6 Hamilton 10 Halifax 11 Kelowna 15 Mississauga 14 Montreal 53
3.13 SOC A sample of 25 first-year students at a university wrote a test that measured their literacy skills in reading and writing (their test scores were the number of incorrect answers, so the higher the score, the poorer the skills).a. Compute the median and mean scores for these data.10 43 30
3.12 SW For the test scores first presented in problem 2.6 and reproduced below, compute a mean and standard deviation for both the pretest and post-test. Based on these statistics, write a paragraph describing how the sample changed from test to test.Case Pretest Post-Test A 8 12 B 7 13 C 10 12 D
3.11 SOC The table below lists the approximate number of cars per 100 population for eight nations.Compute the mean and median for this data. Which is greater in value? Is there a positive skew in this data? How do you know?Nation Number of Cars per 100 Population United States 50 Canada 45 France
3.10 SW As the head of a social services agency, you believe that your staff of 20 social workers is very much overworked compared to 10 years ago. The case loads for each worker are reported below for each of the two years in question. Has the average case load increased? What measure of central
3.9 SOC In problem 3.8, you computed the mean and median weekly earnings for 10 Canadian provinces in two separate years. Now compute the standard deviation for each year. Did the average weekly earnings for the provinces become more or less variable over the period?
3.8 SOC The table below lists the average weekly earnings for the 10 provinces in 2000 and 2005(fictitious data). Compute the mean and median for each year and compare the two measures of central tendency. Which measure of central tendency is greater for each year? Are the distributions skewed? In
3.7 Compute the median, interquartile range, mean, and standard deviation of the 10 scores reported below.10, 12, 15, 20, 25, 30, 32, 35, 40, 50
3.6 SOC Below are four variables for 30 cases taken from a national survey. Age is reported in years.The variable “happiness” consists of answers to the question “Taken all together, would you say that you are (1) very happy, (2) pretty happy, or(3) not too happy?” Respondents were asked
3.5 SOC For 15 respondents, data have been gathered on four variables. Find and report the appropriate measure of central tendency for each variable.
3.4 SOC You have compiled the information below on each of the graduates voted “most likely to succeed” by a local high school for a 10-year period.For each variable, find the appropriate measure of dispersion. (HINT: See Section 3.5 to review the relationship between measure of dispersion and
3.3 PS You have been observing the local Liberal Party in a large city and have compiled some information about a small sample of party regulars.Find the appropriate measure of central tendency for each variable.
3.2 The marital status of residents of four apartment complexes is reported below. Compute the variation ratio (v) for each neighbourhood. Which is the most heterogeneous of the four? Which is the least heterogeneous?
3.1 SOC A variety of information has been gathered from a sample of lower- and upper-year students living in residence at a large university, including their region of birth; the extent to which they support legalization of marijuana (measured on a scale on which 7 5 strong support, 4 5 neutral,
5. Select an appropriate measure of central tendency and dispersion according to level of measurement and skew.
4. Describe and explain the mathematical characteristics of the mean and the standard deviation.
3. Calculate and explain the variation ratio, the range, the interquartile range, and the standard deviation.
2. Calculate and explain the mode, median, and mean.
1. Explain the purposes of measures of central tendency and dispersion and interpret the information they convey.
2.10 PA/CJ As part of an evaluation of the efficiency of your local police force, you have gathered the following data on police response time to calls for assistance during two different years.(Response times were rounded off to whole minutes.) Convert both frequency distributions into percentages
2.9 SOC Twenty-five students completed a questionnaire that measured their attitudes toward interpersonal violence. Respondents who scored high believed that in many situations a person could legitimately use physical force against another person. Respondents who scored low believed that in no
2.8 GER Following are reported the number of times 25 residents of a community for senior citizens left their homes for any reason during the past week.0 2 1 7 3 7 0 2 3 17 14 15 5 0 7 5 21 4 7 6 2 0 10 5 7a. Construct a frequency distribution to display these data.b. What are the midpoints of the
2.7 SOC Sixteen students in their final year of undergraduate studies completed a class to prepare them for the GRE (Graduate Record Examination).Their scores are reported below.420 345 560 650 459 499 500 657 467 480 505 555 480 520 530 589 These same 16 students were given a test of math and
2.6 SW A local youth service agency has begun a sex education program for teenage girls who have been referred by the youth courts. The girls were given a 20-item test for general knowledge about sex, contraception, and anatomy and physiology upon admission to the program and again after completing
2.5 SOC The scores of 15 respondents on four variables are reported below. The numerical codes for the variables are as follows:Sex Support for Legalization of Marijuana Level of Education Age 1 = Male 1 = In favour 0 = Less than high school Actual years 2 = Female 2 = Opposed 1 = High school 2 =
2.4 CJ The numbers of homicides in five states and five Canadian provinces for the years 1997 and 2008 are as follows:1997 2008 State/Province Homicides Population Homicides Population New Jersey 338 8,053,000 376 8,883,000 Iowa 52 2,852,000 76 3,003,000 Alabama 426 4,139,000 353 4,882,000 Texas
2.3 CJ The city of Pearson, Ontario, has a population of 211,732 and experienced 47 bank robberies, 13 murders, and 23 auto thefts during the past year. Compute a rate for each type of crime per 100,000 population. (HINT: Make sure you set up the fraction with size of population in the denominator.)
2.2 At Algebra University, the numbers of males and females in the various major fields of study are as follows:Major Males Females Totals Humanities 117 83 200 Social sciences 97 132 229 Natural sciences 72 20 92 Business 156 139 295 Nursing 3 35 38 Education 30 15 45 Totals 475 424 899 Read each
2.1 SOC The tables that follow report the marital status of 20 respondents in two different apartment complexes. (HINT: Make sure you have the correct numbers in the numerator and denominator before solving the following problems. For example, problem 2.1a asks for “the percentage of respondents
8: Give your table a clear, concise title, and number the table if your report contains more than one. All categories and columns must also be clearly labelled.
7: I nspect the frequency distribution carefully.Has too much detail been lost? If so, reconstruct the table with a greater number of intervals (or use a smaller interval size). Is the table too detailed? If so, reconstruct the table with fewer intervals (or use wider intervals).Are there too many
6: C ount the number of cases in each interval, and report these subtotals in a column labelled“Frequency.” Report the total number of cases(n) at the bottom of this column. The table may also include a column for percentages, cumulative frequencies, and cumulative percentages.
5: S tate the limits of the intervals at the same level of precision as you have used to measure the data. Do not overlap intervals.You will thereby define the intervals so that each case can be sorted into one and only one category.
4: S tate the lowest interval so that its lower limit is equal to or below the lowest score. By the same token, your highest interval will be the one that contains the highest score. Generally, intervals should be equal in size, but unequal and openended intervals may be used when convenient.
3: Find the size of the intervals (i) by dividing R(from step 2) by k (from step 1):i = R /k Round the value of i to a convenient whole number. This will be the interval size or width.
2: Find the range (R ) of the scores by subtracting the low score from the high score.
1: Decide how many intervals (k) you wish to use. One reasonable convention suggests that the number of intervals should be about 10. Many research situations may require fewer than 10 intervals (k = 10), and it is common to find frequency distributions with as many as 15 intervals. Only rarely
4. Construct and analyze bar and pie charts, histograms, and frequency polygons.
3. Construct and analyze frequency distributions for variables at each of the three levels of measurement.
2. Compute and interpret percentages, proportions, ratios, and rates.
1. Explain the purpose of descriptive statistics in making data comprehensible.
1.8 For each research situation below, identify the independent and dependent variables. Classify each in terms of level of measurement and whether or not the variable is discrete or continuous.a. A graduate student is studying sexual harassment on university campuses and asks 500 female students
1.7 For each research situation summarized below, identify the level of measurement of all variables and indicate whether they are discrete or continuous.Also, decide which statistical applications are being used: descriptive statistics (single variable), descriptive statistics (two or more
1.6 For each of the first 20 items in the General Social Survey (see Appendix G), indicate the level of measurement and whether the variable is continuous or discrete.
1.5 Below are brief descriptions of how researchers measured a variable. For each situation, determine the level of measurement of the variable and whether it is continuous or discrete.a. Sex. Respondents were asked to select a category from the following list:_____ Male _____ Femaleb. Honesty.
1.4 Below are some items from a public opinion survey.For each item, indicate the level of measurement and whether the variable will be discrete or continuous. (HINT: Remember that only intervalratio-level variables can be continuous.)a. What is your occupation? _____b. How many years of school
1.3 Distinguish between descriptive and inferential statistics. Describe a research situation that would use both types.
1.2 Find a research article in any social science journal.Choose an article on a subject of interest to you and don’t worry about being able to understand all of the statistics that are reported.a. How much of the article is devoted to statistics per se (as distinct from theory, ideas,
1.1 In your own words, describe the role of statistics in the research process. Using the “wheel of science”as a framework, explain how statistics link theory with research.
4. Identify and describe three levels of measurement and cite examples of variables from each.
3. Distinguish between discrete and continuous variables and cite examples of each.
2. Distinguish between the two general classes of statistical techniques(descriptive and inferential) and identify situations in which each is appropriate.
1. Describe the limited but crucial role of statistics in social research.
47. Consider a situation in which you and a friend both must choose a number from the two numbers 1 and 2. The relevant payoffs are as follows:If you choose 1 and he chooses 1, If you choose 1 and he chooses 2, If you choose 2 and he chooses 1, If you choose 2 and he chooses 2, you both win $10 he
46.Explain how game theory problems (that is, decision-making problems in which there is some “opponent”) can be analyzed using the same techniques which are used in decision-making under uncertainty. Instead of “states of the world,” we have “actions of the opponent.” Might this make
45. Suppose that an investor has a choice of four investments (assume that he must choose one, and only one, of the four—he cannot divide his money among them):A: a savings account in a bank.B: a stock which moves counter to the general stock market.C: a growth stock wThich moves at a faster pace
44. In Exercise 42, find the EVSI and ENGS for samples of size 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, assuming that the cost of sampling is 2. On a graph, draw curves representing EVSI, ENGS, and CS.
43. The payoff in a certain decision-making problem depends on p, the parameter of a Bernoulli process. The payoff functions are State of the World 70R-30B 70B-30R 6 -2-6 10 and P(ai, p) = —10 + lOOp P(a2, p) = 50p.
42. Suppose that the payoff functions of two actions are P (oi, y) = 50 + .5y and P(a2, y) = 70— .5/x, where y is the mean of a normal process with variance 1200.(a) Find the breakeven value, /i&.(b) If the prior distribution is a normal distribution with mean M' — 25 and variance
41. In Exercise 36, show that the EVPI is equal to the expected loss of the optimal action under the prior distribution.
40. Do Exercise 39 with the following payoff table (in dollars):70R-30B Your Guess 70R-30B
39. Consider a bookbag filled with 100 poker chips. You know that either 70 of the chips are red and the remainder blue, or that 70 are blue and the remainder red. You must guess whether the bookbag has 70R-30B or 70B-30R. If you guess correctly, you win $5. If you guess incorrectly, you lose $3.
38. In Exercise 37, suppose that you also wanted to consider samples of other sizes.(a) Find EVSI for a sample of size 2.(b) Find EVSI for a sample of size 5.(c) Find EVSI for a sample of size 10.(d) If the cost of sampling is .50 per unit sampled, find the expected net gain of sampling (ENGS) for
37. In Exercise 36, suppose that sample information is available in the form of a random sample of consumers. For a sample of size one,(a) find the posterior distribution if the one person sampled will purchase the item, and find the value of this sample information(b) find the posterior
36.Astore must decide whether or not to stock a new item. The decision depends on the reaction of consumers to the item, and the payoff table is as follows:
35. In Exercise 8, find the expected value of perfect information. If you could purchase perfect information for $5 (assuming the values in the table are given in terms of dollars), would you do so?
34. In Exercise 12, what is the value of perfect information that there is oil?What is the value of perfect information that there is no oil? From these results, calculate the expected value of perfect information in this example.
33. What is the difference between a terminal decision and a preposterior decision?Are they at all related?
32. Carefully distinguish between “classical” inferential statistics, Bayesian in¬ferential statistics, and decision theory.
31. Compare and contrast the determination of a and (3 in hypothesis testing prob¬lems according to (a) the classical “convention” presented in Chapter 7 and (b) the decision-theoretic approach, as illustrated in Exercise 30.
30. Consider the hypotheses Hq:h = 10 and Hi’.y. = 12, where r is the mean of a normally distributed population with variance 1. The prior distribution is diffuse (take a normal distribution with N' = 0), and a sample of size 10 is to be taken. If L\ = 100 and Ln = 50,(a) find the region of
29. For a sample of size one from a normal population with known variance 25, show that the classical test of Ho-n = 50 versus Hi’./jl — 60 with a — .05 is a likelihood ratio test. That is, show that the rejection region can be expressed in the form LR k, where k is some constant, since you
28.Astatistician is interested in testing the hypotheses and H0:p> 120 H^.ft < 120
27. In Exercise 24, Chapter 8, the marketing manager decides that the loss which will be suffered if the company markets the product and p is in fact only .10 is three times as great as the loss which will be suffered if the company fails to market the product and p is actually .20. Should the
26. In Exercise 11, Chapter 8, the production manager must make an estimate of the mean weight of items turned out by the process in question. His loss function is linear with k0 = ku. What should his estimate be?
25. In Exercise 7, Chapter 8, determine an estimate of p from the posterior dis¬tribution if the loss function is L(a, p) = k(a — p)2.How does this differ from the estimate obtained from the prior distribution using the same loss function? From the estimate obtained from the sample alone?
24. If the loss function in a point estimation problem is of the form L (a, 6)0 1if | a — 9 | < k, otherwise, where k is some very small positive number, what would the optimal estimate be? Can you think of any realistic situations in which the loss function might be of this form?
23. If a statistician wishes to estimate a parameter d subject to a loss function which is linear with k0 = 2kU) and his distribution of 6 is an exponential distribution with X = 4, what is the optimal estimate of 0?
22. If a person faces a point estimation problem with a linear loss function with ku — 4 and k0 = 3, does he need to assess an entire probability distribution or can he determine a certainty equivalent? Explain.
21. Comment on the following statement: “In taking the sample mean as an estimator of the population mean, the statistician is acting essentially as though he had a quadratic loss function.” Explain the difference between the approach to point estimation taken in Chapter 6 and the approach
20. In Exercises 15, 16, 17, and 19, what would the optimal estimates be if the loss function were quadratic instead of linear [that is, if the loss function were of the form (9.13.1)]?
19. An economist is asked to predict a future value of a particular economic indicator, and he thinks that his loss function is linear with ku = 4k0. If his distribution is a uniform distribution on the interval from 650 to 680, what should his estimate be?
18. Prove that if a decision maker’s loss function in a point estimation problem is given by Equation (9.12.1), then the kj (ku + k0) fractile of the decision maker’s distribution of 6 is the optimal point estimate.
17. A sales manager is asked to forecast the total sales of his division for a forth¬coming period of time. He feels that his loss function is linear as a function of the difference between his estimate and the true value, but he also feels that an error of overestimation is three times as serious
16. A hot dog vendor at a football game must decide in advance how many hot dogs to order. He makes a profit of $.10 on each hot dog which is sold, and he suffers a $.20 loss on hot dogs which are unsold. If his distribution of the number of hot dogs that will be demanded at the football game is a
15. Suppose that a contractor must decide whether or not to build any speculative houses (that is, houses that he would have to find a buyer for), and if so,
14. Attempt to determine your own utility function for money in the range from —$500 to +$500. If you were given actual decision-making situations, would you act in accordance with this utility function?
13. Comment on the following statement from the text: “Some individuals appear to be risk-takers for some decisions (such as gambling) and risk-avoiders for other decisions (such as purchasing insurance).” Can you explain why this phenomenon occurs? Can you draw a utility function which would
12. Suppose that you are contemplating drilling an oil well, with the following payoff table (in terms of thousands of dollars):

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Statistics Probability Inference And Decision 1st Edition Robert L. Winkler, William L. Hays - Solutions

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