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nonparametric statistical inference
Fundamentals Of Statistical Reasoning In Education 3rd Edition Theodore Coladarci, Casey D. Cobb, Edward W. Minium, Robert C. Clarke - Solutions
20. Suppose you randomly select two students from the group in Problem 18. What is the probability that:
*19. Is the probability in Problem 18d a one- or two-tailed probability? (Explain.)
(g) The probability is .10 that the student’s SAT-CR score will fall below an SAT-CR score of ____?
(f) The probability is .10 that the student’s SAT-CR score will fall above an SAT-CR score of ____?
(e) The probability is .50 that the student’s SAT-CR score will be between what two values?
*18. The verbal subscale on the SAT (SAT-CR) has a normal distribution with a mean of 500 and a standard deviation of 100. Consider the roughly one million high school seniors who took the SAT last year. If one of these students is selected at random, what is the probability that his or her SAT-CR
17. You’re back on the slot machine from Problem 13. What is the probability that:(a) an orange will appear on exactly two of the wheels?(b) an orange will appear on at least two of the wheels?(c) a jackpot label will appear on at least one of the wheels?
16. Your statistics instructor administers a test having five multiple-choice items with four options each. List the ways in which one can guess correctly on exactly four items on this test.What is the probability of:(a) guessing correctly on any one of the five items?(b) guessing incorrectly on
(e) What is the probability of guessing correctly on at least two of the three items?
(d) List the way(s) you can guess correctly on all three of the items.
(c) What is the probability of guessing correctly on exactly two out of the three items?
(b) What is the probability of guessing correctly on the first two items and guessing incorrectly on the third item?
(a) List the way(s) you can guess correctly on exactly two out of the three items.
*15. You make random guesses on three consecutive true–false items.
14. Suppose you pull the lever on the slot machine described in Problem 13. What is the probability that:(a) either an orange or a lemon or a jackpot label will appear on the middle wheel?(b) a jackpot label will appear on all three wheels?(c) cherries will appear on all three wheels?
*13. A slot machine has three wheels that rotate independently. When the lever is pulled, the wheels rotate and then come to a stop, one by one, in random positions. The circumference of each wheel is divided into 25 equal parts and contains four pictures each of six different fruits and one
12. Events A and B are mutually exclusive. Can they also be independent? (Explain.)
11. For each of the instances described in Problem 10, indicate whether the events are independent.
(i) Being 15 years old and voting (legally) in the last national election.
(h) Winning on one play of a slot machine and winning on the very next play.
(g) Obtaining a 3 and an even number on a single roll of a die.
(f) Disliking rock music and attending a rock concert.
(e) Obtaining three aces in two consecutive hands dealt each time from a complete, well-shuffled deck of playing cards.
(d) Obtaining a final grade of A and obtaining a final grade of C for your first course in statistics.
(c) As an undergraduate student, being an education major and being a psychology major.
(b) Being a male and being pregnant.
(a) Obtaining heads on the first toss of a coin and tails on the second toss.
*10. In which of the following instances are the events mutually exclusive?
(b) What is the probability of an even number on the first die and a 3 on the second?
(a) What is the probability of an even number or a 3 on the first die?
*9. Two fair dice are rolled.
8. What is the distinction, if any, between a relative frequency distribution and a probability distribution? (Explain.)
7. Suppose you make three consecutive random selections from the group of 200 students in Problem 5. After each selection, you record the grade and sex of the student selected and replace him or her back in the group before making your next selection. First, determine the following three
*6. Because a grade of F is one of five possible letter grades, why isn’t 1/5, or .20, the answer to Problem 5a?
Using the basic rule given in Section 9.3, determine the probability of selecting:(a) an F student(b) a female(c) a female B student(d) a male with a grade below C Exercises 187
5. A student is selected at random from the group of 200 represented in the table below.Sex of Student Course Grade Male Female Total A 18 12 30 B 30 30 60 C 53 27 80 D 12 8 20 F 7 3 10 f 120 80 200
*4. The following question is asked on a statistics quiz: If one person is selected at random out of a large group, what is the probability that he or she will have been born in the month beginning with the letter J? Jack Sprat reasons that because three of the 12 months begin with the letter J,
*3. Six CD players, three wide-screen TVs, and one laptop are given out as door prizes at a local club. Winners are determined randomly by the number appearing on the patron’s admission ticket. Suppose 300 tickets are sold (and there are no no-shows). What is the probability that a particular
(b) What is the probability of getting tails on the sixth toss?
(a) Is it therefore more likely that you will get tails on the sixth toss? (Explain.)
2. Imagine that you toss an unbiased coin five times in a row and heads turns up every time.
*1. In an education experiment, a group of students is randomly divided into two groups.The two groups then receive different instructional treatments and are observed for differences in achievement. Why would the researcher feel it necessary to apply \statistical inference" procedures to the
4. What is the probability of an event as extreme as three heads?This question similarly involves both sides of the probability distribution, insofar as no heads is just as \extreme" as four heads, and one head is just as extreme as three heads. (That is, \as rare as" and \as extreme as" are
3. What is the probability of an event as rare as four heads?To determine this probability, first you must acknowledge that obtaining four heads is just as are" as obtaining no heads, as the symmetry of Figure 9.1 testifies. Thus, both sides of the probability distribution are implicated by the
2. What is the probability of obtaining no more than one head?The reasoning is similar here, although now you are on the other side of the probability distribution. Because either no heads or one head satisfies this condition, the probability of obtaining no more than one head is p0heads þ p1head
. What is the probability of obtaining at least three heads?The condition of \at least three heads" is satisfied if you obtain either three heads or four heads. The probability of obtaining at least three heads is therefore p3heads þ p4heads ¼ :25 þ :0625 ¼ :3125:
17. At the end of the section on \setting up the margin of error," we asked if you can see from Table A in Appendix C how we got \1.00" and \2.58" for 68% and 99% confidence, respectively. Can you?
(d) As accurately as possible, draw in the new regression line using the figure below(from which Student 26 has been deleted). How does the result compare with your response to Problems 16a and 16b?
(c) With Student 26 removed, the relevant summary statistics are X ¼ 69:45, SX ¼ 9:68, Y ¼ 100:83, SY ¼ 14:38, r ¼ :79. Calculate the new intercept and slope.
(b) How should the removal of Student 26 affect the magnitude of the intercept? the slope?
(a) Imagine you deleted this case, recalculated intercept and slope, and drew in the new regression line. Where do you think the new line would lie relative to the original regression line?Why? (Refer to the least-squares criterion.)
16. At the end of Section 8.3, we asked you to consider how the location of Student 26 would affect the placement of the regression line in Figure 8.4.
15. Consider the situation described in Problem 13. By embarking on a new but very expensive testing program, Ecalpon Tech can improve the correlation between the aptitude score and GPA to r ¼þ:55. Suppose the primary concern is the accuracy with which GPAs of individuals can be predicted. Would
(c) What is an excellent way to check and see whether the assumptions are being appreciably violated?
(b) Explain the role of each assumption underlying the procedures used to answer Problems 13d–13g.
14. (a) What assumption(s) underlie the procedure used to answer Problem 13b?
(g) For students with aptitude scores the same as Mike’s, what proportion would be expected to obtain a GPA of 2.5 or better?
(f) For students with aptitude scores the same as Val’s, what proportion would be expected to obtain a GPA of 2.0 or below?
(e) For students with aptitude scores the same as Val’s, what proportion would you expect to obtain a GPA better than the first-year mean?
(d) Set up the 95% confidence limits around Val’s and Mike’s predicted GPAs.
(c) Compute the standard error of estimate.
(b) Val and Mike score 485 and 710, respectively, on the aptitude test. Predict their first-year GPAs.
(a) Calculate the raw-score intercept and slope; state the regression equation.
(f) Show how Formula (8.5) illustrates this principle.*13. The following data are for first-year students at Ecalpon Tech:Aptitude Score First-year GPA X ¼ 560:00 Y ¼ 2:65 SX ¼ 75:00 SY ¼ :35 r ¼þ:50
(e) State the principle that emerges from your answers to Problems 12a to 12d.
12. (No calculations are necessary for this problem.) Suppose the following summary statistics are obtained from a large group of individuals: X ¼ 52:0, SX ¼ 8:7, Y ¼ 147:3, SY ¼ 16:9. Dorothy receives an X score of 52. What is her predicted Y score if:(a) r = 0?(b) r = .55?(c) r = +.38?(d) r
8.170 Chapter 8 Regression and Prediction
(c) Convert the predicted z scores from Problem 11b back to predicted heights in inches and compare with the results of Problem 3b.
(b) Use the standard-score form of the regression equation to obtain their predicted z scores for height as adults.
(a) Convert to z scores the 10-year-old heights of Jean, Albert, and Burrhus.
*11. Consider the situation described in Problem 3.
10. For each condition in Problem 9, state the regression equation in z-score form.
*9. Gayle falls one standard deviation above the mean of X. What is the correlation between X and Y if her predicted score on Y falls:(a) one standard deviation above?(b) one-third of a standard deviation below?(c) three-quarters of a standard deviation above?(d) one-fifth of a standard deviation
(b) Using a straightedge with the regression line, estimate (accurate to one decimal place) the predicted principal ratings for the three applicants in Problem 6c. Compare these values with the Y0 values you calculated earlier from the regression equation.
6. Mark off divisions on the two axes so that the plot will be as large as possible and as close to square as possible. Plot the data points accordingly, and draw in the regression line as described in Section 8.3.
8. (a) On an 8½@11@ piece of graph paper, construct a scatterplot for the data of Problem
*7. Suppose X in Problem 6 were changed so that there is absolutely no relationship between test scores and principal ratings (r ¼ 0).(a) What would be the predicted rating for each of the three applicants? (Explain.)(b) What would be the intercept and slope of the regression equation for
(d) If in fact these data were real, what objections would you have to using the equation from Problem 6b for prediction in a real-life situation?
(c) Suppose that three teachers apply for positions in this school, obtaining scores of 18, 32, and 42, respectively, on the teacher certification test. Compute their predicted ratings of teacher effectiveness.
(b) Using values from Problem 6a, calculate the intercept and slope; state the regression equation.
(a) Compute the summary statistics required for determining the regression equation for predicting principal ratings from teacher certification test scores.
6. Following are the scores on a teacher certification test administered prior to hiring (X)and the principal’s ratings of teacher effectiveness after three months on the job (Y) for a group of six first-year teachers (A–F):A B C D E F Test score (X): 14 24 21 38 34 49 Principal rating (Y): 7 4
*5. Interpret the slope from Problems 3 and 4.Exercises 169
(d) Compute the sum of the prediction errors for these five individuals, and state the generalization that this sum illustrates (within the limits of rounding error).
(c) Compute the mean of the predicted scores and compare with the summary statistics above. What important generalization (within the limits of rounding error) emerges from this comparison?
(b) Obtain predicted scores for Keith, Bill, Charlie, Brian, and Mick. Compare your answers with those obtained in Problem 1a; explain any discrepancies.
(a) From these values, compute intercept and slope for the regression equation;state the regression equation.
*4. The following are the summary statistics for the scores given in Problem 1:X ¼ 5:00; SX ¼ 2:97; Y ¼ 6:00; SY ¼ 3:03; r ¼þ:62
(c) Consider Jean’s predicted adult height. In what sense is that value a mean?
(b) With this regression equation, predict the adult height for the following 10-year-olds:Jean P. (42.5 in.), Albert B. (55.3 in.), and Burrhus S. (50.1 in.).
(a) Use the values above to compute intercept and slope for predicting adult height from height at age 10 (round to the second decimal place); state the regression equation, using the form of Formula (8.1).
*3. A physical education teacher, as part of a master’s thesis, obtained data on a sizable sample of males for whom heights both at age 10 and as adults were known. The following are the summary statistics for this sample:Height at Age 10 Adult Height X ¼ 48:3 Y ¼ 67:3 SX ¼ 3:1 SY ¼ 4:1 r
(b) Why is it an estimated rather than an actual mean?
(a) In what sense can the value 550 be considered an estimated mean?
2. The relationship between student performance on a state-mandated test administered in the fourth grade and again in the eighth grade has been analyzed for a large group of students in the state. Ellen obtains a score of 540 on the fourth-grade test. From this, her performance on the eighth-grade
(d) If any other line were used for prediction, how would the error sum of squares compare with your answer to Problem 1c?
(c) Use the answers from Problem 1b to compute the error sum of squares.
(b) Use the answers from Problem 1a to determine the error in prediction for each student.
(a) Use a straightedge with the regression line to estimate (to one decimal place) the predicted Y score (Y0) of each student.
1. The scatterplot and least-squares regression line for predicting Y from X is given in the figure below for the following pairs of scores from a pretest and posttest:Keith Bill Charlie Brian Mick Pretest (X) 8 9 4 2 2 Posttest (Y) 10 6 8 5 1 prediction correlation and prediction regression line
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