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nonparametric statistical inference

Fundamentals Of Statistical Reasoning In Education 4th Edition Theodore Coladarci, Casey D. Cobb - Solutions

13. Suppose you don’t know anything about the shape of the population distribution of ratings used in Problems 11 and 12. Would this lack of knowledge have any implications for solving Problem 11? Problem 12? (Explain.)
12. Repeat Problem 11h using a sample of size 100.(a) What is the effect of this larger sample on the standard error of the mean?(b) What is the effect of this larger sample on the limits within which the central 95% of sample means fall?(c) Can you see an advantage of using large samples in
11. A population of peer ratings of physical attractiveness is approximately normal withμ 5 2 and σ 1 6. A random sample of four ratings is selected from this population.(a) Calculate σX .What is the probability of obtaining a sample mean:(b) above 6.6?(c) as extreme as 3.8?(d) below 4.4?(e)
10. Suppose you collected an unlimited number of random samples of size 36 from the 10.population in Problem 9.(a) What would be the mean of the resulting sample means?(b) What would be the standard deviation of the sample means?(c) What would be the shape of the distribution of sample means? (How
9. Given: 9. μ 100 and σ 30 for a normally distributed population of observations. Suppose you randomly selected from this population a sample of size 36.(a) Calculate the standard error of the mean.(b) What is the probability that the sample mean will fall above 92?(c) What is the probability
8. What are the key questions to be answered in any statistical inference problem? 8.
7. What are the three defining characteristics of any sampling distribution of means?
6. Explain on an intuitive basis why the sampling distribution of means for n 2 selected from the “flat” distribution of Figure 10.4a has more cases in the middle than at the extremes. (Hint: Compare the number of ways an extremely high or an extremely low mean could be obtained with the number
5. Suppose you did not know Formula (10.2) for 5. σX . If you had unlimited time and resources, how would you go about obtaining an empirical estimate of σX for samples of three cases each drawn from the population of Problem 4?
4. A certain population of observations is bimodal (see Figure 3.10b).(a) Suppose you want to obtain a fairly accurate picture of the sampling distribution of means for random samples of size 3 drawn from this population. Suppose also that you have unlimited time and resources. Describe how,
3. A researcher conducting a study on attitudes toward 3. “homeschooling” has her assistant select a random sample of 10 members from a large suburban church. The sample selected comprises nine women and one man. Upon seeing the uneven distribution of sexes in the sample, the assistant
2. After considering the sampling problems associated with Problem 1, your friend decides to interview people who literally are “on the street.” That is, he stands on a downtown sidewalk and takes as his population passersby who come near enough that he might buttonhole them for an interview.
1. “The average person on the street is not happy,” or so claimed the newscaster after interviewing patrons of a local sports bar regarding severe sanctions that had been imposed on the state university for NCAA infractions.(a) What population does the newscaster appear to have in mind?(b) What
20. Suppose you randomly select two students from the group in Problem 18. What is the probability that:(a) the first student falls at least 100 SAT-CR points away from the mean (in either direction)?(b) both students obtain SAT-CR scores above 700?(c) the first student obtains a score above 650
19. Is the probability in Problem 18d a one- or two-tailed probability? (Explain.) 19.
18. The verbal subscale on the SAT (SAT-CR) has a normal distribution with a mean of 500 18.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
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
15. You make random guesses on three consecutive true 15. –false items.(a) List the way(s) you can guess 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?
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 13.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.
10. In which of the following instances are the events 10. mutually exclusive?(a) Obtaining heads on the first toss of a coin and tails on the second toss.(b) Being a male and being pregnant.(c) As an undergraduate student, being an education major and being a psychology major.(d) Obtaining a final
9. Two fair dice are rolled. 9.(a) What is the probability of an even number or a 3 on the first die?(b) What is the probability of an even number on the first die and a 3 on the second?
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 6. five possible letter grades, why isn’t 1/5, or .20, the answer to Problem 5a?
5. A student is selected at random from the group of 200 represented in the table below.
4. The following question is asked on a statistics quiz: 4. 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 iPads, three 3. flat-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 patron
2. Imagine that you toss an unbiased coin five times in a row and heads turns up every time.(a) Is it therefore more likely that you will get tails on the sixth toss? (Explain.)(b) What is the probability of getting tails on the sixth toss?
1. In an education experiment, a group of students is randomly divided into two groups. 1.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
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?
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.(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
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 55. Suppose the primary concern is the accuracy with which GPAs of individuals can be predicted. Would the
14. (a) What assumption(s) underlie the procedure used to answer Problem 13b?(b) Explain the role of each assumption underlying the procedures used to answer Problems 13d–13g.(c) What is an excellent way to check and see whether the assumptions are being appreciably violated?
13. The following data are for 13. first-year students at Ecalpon Tech:Aptitude Score First-year GPA X 560 00 Y 2 65 SX 75 00 SY 35 r 50(a) Calculate the raw-score intercept and slope; state the regression equation.(b) Val and Mike score 485 and 710, respectively, on the aptitude test. Predict
12. (No calculations are necessary for this problem.) Suppose the following summary statistics are obtained from a large group of individuals: X 52 0, SX 8 7, Y 147 3, SY 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 −1.00?(e) State
11. Consider the situation described in Problem 3. 11.(a) Convert to z scores the 10-year-old heights of Jean, Albert, and Burrhus.(b) Use the standard-score form of the regression equation to obtain their predicted z scores for height as adults.(c) Convert the predicted z scores from Problem 11b
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 9. 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
8. (a) On an 8½ × 11 piece of graph paper, construct a scatterplot for the data of Problem 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
7. Suppose 7. X in Problem 6 were changed so that there is absolutely no relationship between test scores and principal ratings (r 0).
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):ABCDEF Test score (X): 14 24 21 38 34 49 Principal rating (Y): 7 4 10 8
5. Interpret the slope from Problems 3 and 4. 5.
4. The following are the summary statistics for the scores given in Problem 1: 4.X 5 00 SX 2 97 Y 6 00 SY 3 03 r 62(a) From these values, compute intercept and slope for the regression equation; state the regression equation.(b) Obtain predicted scores for Keith, Bill, Charlie, Brian, and Mick.
3. A physical education teacher, as part of a master 3. ’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:(a) Use the values above to compute intercept and slope for predicting
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
1. The scatterplot and least-squares regression line for predicting 1. Y from X is given in the figure below for the following pairs of scores from a pretest and posttest:(a) Use a straightedge with the regression line to estimate (to one decimal place) the predicted Y score (Yˆ ) of each
16. Consider the situation where there is absolutely 16. no variability in Y.(a) What would be the standard deviation of Y?(b) What would be the covariance between X and Y?(c) What would be the Pearson r? (Don’t respond reflexively!
15. Some studies have found a strong negative correlation between how much parents help their children with homework (X) and student achievement (Y). That is, children who receive more parental help on their homework tend to have lower achievement than kids who receive little or no parental help.
14. It is common to 14. find that the correlation between aviation aptitude test scores (X) and pilot proficiency (Y) is higher among aviation cadets than among experienced pilots.How would you explain this?
13. Does a low r necessarily mean that there is little “association” between two variables?(Explain.)
12. An 12. r of .60 was obtained between IQ (X) and number correct on a word-recognition test (Y) in a large sample of adults. For each of the following, indicate whether or not r would be affected, and if so, how (treat each modification as independent of the others):(a) Y is changed to number of
11. For a particular set of scores, SX 3 and SY 5. What is the largest possible value of the covariance? (Remember that r can be positive or negative.)
10. r − 47, SX 6, and SY 4. What is the covariance between X and Y?
9. The covariance between X and Y is −72, SX 8 and SY 11. What is the value of r?
8. r from Problem 7.
7. Suppose you change the data in Problem 3a 7. so that the bottom case is X 1 and Y 12 rather than X 1 and Y 2.(a) Without doing any calculations, state how (and why) this change would affect the numerator of the covariance and, in turn, the covariance itself.
6. (a) Using the data in Problem 3, divide each value of X by 2 and construct a scatterplot showing the relationship between X and Y.(a) How do your impressions of the new scatterplot compare with your impressions of the original plot?(b) What is the covariance between X and Y?(c) How is the
5. What is the covariance for the data in Problem 3 5. ?
4. (a) Using the data in Problem 3, determine r from both the defining formula and the calculating formula.(b) Interpret r within the context of the coefficient of determination.
3. (a) Prepare a scatterplot for the data below, following the guidelines presented in this chapter.
2. Why is it important to inspect scatterplots?
1. Give examples, other than those mentioned in this chapter, of pairs of variables you would expect to show:(a) a positive association(b) a negative association(c) no association at all
16. Consider the effect sizes you computed for Problem 15 of Chapter 5. Interpret these 16.within the context of area under the normal curve, as discussed in Section 6.9.
15. X 20 and S 5 on a test of mathematics problem solving (scores reflect the number of problems solved correctly). Which represents the greatest difference in problem-solving ability: P5 vs. P25, or P45 vs. P65? Why? (Assume a normal distribution.)
14. The mean of a set of 14. z scores is always zero. Does this suggest that half of a set of z scores will always be negative and half always positive? (Explain.)
13. The following 13. five scores were all determined from the same raw score distribution(assume a normal distribution with X 35 and S 6). Order these scores from best to worst in terms of the underlying level of performance.(a) percentile rank 84(b) X 23(c) deviation score 0(d) T 25(e) z 1 85
12. Convert each of the scores in Problem 2 to T scores.
11. Given a normal distribution with 11. X 500 and S 100, find the percentile ranks for scores of:(a) 400(b) 450
10. Given a normal distribution of tests scores, with X 250 and S 50:(a) What score separates the upper 30% of the cases from the lower 70%?(b) What score is the 70th percentile (P70)?(c) What score corresponds to the 40th percentile (P40)?(d) Between what two scores do the central 80% of scores
9. In a normal distribution, what is the z score:(a) above which the top 5% of the cases fall?(b) above which the top 1% of the cases fall?(c) below which the bottom 5% of the cases fall?(d) below which the bottom 75% of the cases fall?
8. In a normal distribution, what 8. z scores:(a) enclose the middle 99% of cases?(b) enclose the middle 95% of cases?(c) enclose the middle 75% of cases?(d) enclose the middle 50% of cases?
7. In a normal distribution, what proportion of cases fall: 7.(a) outside the limits z −1 00 and z 1 00?(b) outside the limits z − 50 and z 50?(c) outside the limits z −1 26 and z 1 83?(d) outside the limits z −1 96 and z 1 96?
6. In a normal distribution, what proportion of cases fall between:(a) z −1 00 and z 1 00?(b) z −1 50 and z 1 50?(c) z −2 28 and z 0?(d) z 0 and z 50?(e) z 75 and z 1 25?(f) z − 80 and z −1 60?
5. In a normal distribution, what proportion of cases fall (report to four decimal places): 5.(a) above z 1 00?(b) below z −2 00?(c) above z 3 00?(d) below z 0?(e) above z −1 28?(f) below z −1 62?
4. Make a careful sketch of the normal curve. For each of the z scores of Problem 3, pinpoint as accurately as you can its location on that distribution.
3. Convert the following z scores back to academic self-concept scores from the distribution of Problem 2 (round answers to the nearest whole number):(a) 0(b) −2.10(c) 1.82(d) −.75(e) .25(f) 3.10
2. X 82 and S 12 for the distribution of scores from an “academic self-concept”instrument that is completed by a large group of elementary-level students (high scores reflect a positive academic self-concept). Convert each of the following scores to a z score:(a) 70(b) 90(c) 106(d) 100(e) 62(f)
1. What are the various properties of the normal curve?
15. Imagine you obtained the following results in an investigation of sex differences among 15.high school students:Mathematics Achievement Verbal Ability Male n 32 Female n 34 Male n 32 Female n 34 XM 48 SM 9 0 XF 46 SF 9 2 XM 75 SM 12 9 XF 78 SF 13 2(a) What is the pooled standard deviation for
14. The mean is 67 for a large group of students in a college physics class; Duane obtains a 14.score of 73.(a) From this information only, how would you describe his performance?(b) Suppose S 20. Now how would you describe his performance?(c) Suppose S 2. Now how would you describe his performance?
13. Given: X 500 and S 100 for the SAT-CR, what percentage of scores would you expect to fall(a) between 400 and 600?(b) between 300 and 700?(c) between 200 and 800?
12. Determine the sum of squares SS corresponding to each of the following standard deviations n 30 :(a) 12(b) 9(c) 6(d) 4.5
11.8 8 8 8 8 6 6 8 10 10 4 6 8 10 12 1004 1006 1008 1010 1012(a) Upon inspection, which show(s) the least variability? the most variability?(b) For each set of scores, compute the mean; compute the variance and standard deviation directly from the deviation scores.(c) What do the results of Problem
10. Imagine that each of the following pairs of means and standard deviations was 10.determined from scores on a 50-item test. With only this information, describe the probable shape of each distribution. (Assume a normal distribution unless you believe the information presented suggests
9. Why is the variance little used as a descriptive measure?
8. After you have computed the mean, median, range, and standard deviation of a set of 8.40 scores, you discover that the lowest score is in error and should be even lower.Which of the statistics above will be affected by the correction? (Explain.)
7. If you wanted to decrease variance by adding a point to some (but not all) scores in a distribution, which scores would you modify? What would you do if you wanted to increase variance?
6. For each of the following statistics, what would be the effect of adding one point to 6.every score in a distribution? What generalization do you make from this? (Do this without calculations.)(a) mode(b) median(c) mean(d) range(e) variance(f) standard deviation
5. Given: S2 18 and SS 900. What is n?
4. Determine the standard deviation for the following set of scores. X: 2.5, 6.9, 3.8, 9.3, 5.1, 8.0.
3. For each set of scores below, compute the range, variance, and standard deviation. 3.(a) 3, 8, 2, 6, 0, 5(b) 5, 1, 9, 8, 3, 4(c) 6, 4, 10, 6, 7, 3
2. Each of five raw scores is converted to a deviation score. The values for four of the deviation scores are as follows: −4, 2, 3, −6. What is the value of the remaining deviation score?
1. Give three examples, other than those mentioned in this chapter, of an “average”(unaccompanied by a measure of variability) that is either insufficient or downright misleading. For each example, explain why a variability measure is necessary.
16. If the eventual purpose of a study involves statistical inference, which measure of central tendency is preferable (all other things being equal)? (Explain.)
15. From an article in a local newspaper: “The median price for the houses sold was$125,000. Included in the upper half [of houses sold] are the one or two homes that could sell for more than $1 million, which brings up the median price for the entire market.” Comment?
14. Which measure(s) of central tendency would you be unable to determine from the 14.following data? Why?

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