Fundamentals Of Statistical Reasoning In Education 3rd Edition Theodore Coladarci, Casey D. Cobb, Edward W. Minium, Robert C. Clarke - Solutions

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13. Where must the mode lie in the distribution of GPAs in Table 2.5?
12. What is the mean, median, and mode for the distribution of scores in Table 2.2? 12.
11. Suppose you were a school psychologist and were interested only in improving the 11. median self-esteem score of children in your school. On which of the following students would you work the hardest: (1) those with the lowest self-esteem scores, (2) those with the highest, (3) those just below
10. X 23, Mdn 28, mode 31 for a particular distribution of 25 scores. It was subsequently found that a scoring mistake had been made: one score of 43 should have been a 34.(a) What is the correct value for X ?(b) How would the Mdn and mode be affected by this error?
9. At a local K–6 school, the four K–2 teachers have a mean of 15 students per class, while the five teachers for grades 3–6 have a mean of 18 students per class. What is the mean number of students across the nine teachers in this school?
8. A newspaper editor once claimed that more than half of American families earned a below-average income. Could this claim possibly be correct? (Explain.)
7. State the likely relative positions of the mean, median, and mode for the following 7.distributions:(a) family income in a large city(b) scores on a very easy exam(c) heights of a large group of 25-year-old males(d) the number of classes skipped during the year for a large group of undergraduate
6. Comment on the probable shape for each of the following distributions (knowing noth- 6.ing else about these distributions):(a) X 52, Mdn 55, mode 60(b) X 79, Mdn 78, mode 78(c) X 50, Mdn 50, mode 60 40(d) X 28, Mdn 26, mode 20
5. (a) What is meant by the “balance point” of a distribution of scores? How is the expression, Σ X − X 0, relevant to this concept?(b) Show that Σ X − X 0 for the following sample of scores: 2, 5, 7, 8, 13.
4. In the following quotation, taken verbatim from a company newsletter, the author was attempting to provide statistical enlightenment:One of the most misused words is the word “average.” It is often confused with“mean.” The difference is this: If five products sell for $2, $3, $5, $8, and
3. Which measure of central tendency is most easily estimated from a histogram or frequency polygon? Why?
2. For each of the following sets of scores, 2. find the mode, the median, and the mean:(a) 12, 10, 8, 22, 8(b) 14, 12, 25, 17(c) 10, 6, 11, 15, 11, 13
1. List the physical characteristic of a frequency curve that corresponds to each of the three measures of central tendency.
3. Following the guidelines presented in this chapter, construct a histogram (using fre- 4.quencies) to exhibit the distribution of test scores obtained in Problem 7 of Chapter 2.(Be sure to provide clear labels for the two axes and a title.)Suppose in Problem 4 you had used percentages along the
2. Describe the similarities and differences between a bar chart and a histogram.* Give the midpoints of each of the following intervals (assume the scores are rounded to 3.the nearest whole number or decimal place indicated unless otherwise specified):(a) 10–14(b) 200–399(c) 2.50–2.74(d)
1. Why might a statistically knowledgeable person prefer to inspect a frequency distribu- 1.tion rather than a graph? What would be an argument against this position?
13 using an interval width of 5 and 15–19 as the lowest interval. Compare the result with the distribution obtained in Problem 13. From this example, what would you say are the advantages of sometimes using a larger interval size and thus fewer intervals?
14. Construct a relative frequency distribution (use proportions) from the scores in Problem 14.
13. The following scores were obtained by middle-level students on an “educational aspirations” assessment:41 33 18 41 36 50 27 34 36 36 36 36 39 33 40 48 29 41 28 39 30 44 41 39 45 30 36 27 21 46 40 47 46 47 35 24 32 46 33 39 33 45 39 31 37 46 34 18 30 35 27 42 27 31 33 44 39 36 24 27 30 24 22
12. Imagine the data below are the GPAs for a sample of 60 sophomores at your university. 12.Prepare a relative frequency distribution (use proportions), using an interval width of.30 and .90–1.19 as the score limits for the lowest interval.3.08 1.81 3.63 2.52 2.97 3.48 1.00 2.70 2.95 3.29 1.40
11. Provide the exact limits for the data we presented earlier in Table 2.5:GPA f Exact Limits 3.80–3.99 2 3.60–3.79 3 3.40–3.59 4 3.20–3.39 6 3.00–3.19 5 2.80–2.99 9 2.60–2.79 7 2.40–2.59 2 2.20–2.39 3 2.00–2.19 3 1.80–1.99 1 1.60–1.79 1
10. Construct a frequency distribution for these data.Imagine that you want to compare the frequency distributions for males and females 10.separately (on some variable), and there are considerably more females than males.Which would you concentrate on—the original frequencies or the relative
9. Suppose that the racial/ethnic breakdown of participants in your investigation is as follows: African American, n 25; White, n 90; Asian, n 42; Hispanic, n 15; “other,”
8. The following is the cumulative frequency distribution for 30 scores on a “test anxiety”survey.(a) Fill in the three blank columns (round the cumulative percentages to the nearest whole number).(b) Find the cumulative percentage and percentile ranks for each of the following scores: 67, 57,
7. Thirty prospective teachers take a standards-based teacher competency test. The 7.results are as follows (each score reflects the percentage of standards for which the prospective teacher demonstrates proficiency):81 91 89 81 79 82 70 92 80 64 73 86 87 72 74 75 90 85 83 82 79 82 78 96 77 85 83
6. Convert the following percents to proportions (again, to the same degree of accuracy):(a) 42%(b) 6.6%(c) 43.7%(d) 78%(e) .8%
5. Convert the following proportions to percents (to the same degree of accuracy): 5.(a) .26(b) .05(c) .004(d) .555(e) .79
4. For each of the following intervals, give (1) the interval width, (2) the exact limits of the interval, and (3) the score limits and exact limits of the next higher interval (assume the scores are rounded to the nearest whole number or decimal place indicated unless otherwise specified):(a)
3. The lowest and highest scores are given below for different sets of scores. In each case, 3.the scores are to be grouped into class intervals. For each, give (1) the range, (2) your choice of class interval width, (3) the score limits for the lowest interval, and (4) the score limits for the
2. Comment on the following statement: “The rules for constructing frequency distributions have been carefully developed and should be strictly adhered to.”
1. List the objectionable features of this set of clas 1. s intervals (score limits) for a hypothetical frequency distribution:Score Limits 25–30 30–40 40–45 50–60 60–65
5. In his travels, one of the authors once came upon a backroad sign announcing that a small town was just around the corner. The sign included the town’s name, along with these facts:Population 562 Feet above sea level 2150 Established 1951 TOTAL 4663 Drawing on what you have learned in this
4. Round the following numbers as speci 4. fied (review Appendix A.7 if necessary):(a) to the nearest whole number: 8.545, −43.2, 123.01, .095(b) to the nearest tenth: 27.33, 1.9288, −.38, 4.9746(c) to the nearest hundredth: −31.519, 76.0048, .82951, 40.7442
3. For the three questions that follow, illustrate your reasoning with a variable from the list in Problem 1.(a) Can a ratio variable be reduced to an ordinal variable?(b) Can an ordinal variable be promoted to a ratio variable?(c) Can an ordinal variable be reduced to a nominal variable?
2. Which of the variables from Problem 1 are qualitative variables and which are quantitative variables?
1. Indicate which scale of measurement each of the following variables re 1. flects:(a) the distance one can throw a shotput(b) urbanicity (where 1 urban, 2 suburban, 3 rural)(c) school locker numbers(d) SAT score(e) type of extracurricular activity (e.g., debate team, field hockey, dance)(f)
(e) The keyboarding speed of secretaries at her company is lower than the national average.
(d) H0 is rejected because the sample z falls in the critical region (similarly, p a).
5. (a) 1.00(b) 22.00(c) (:0228)(2)5:05
4. (a) 1.67(b) 21.20(c) (:1151)(2)5:23(d) H0 is retained because the sample z falls short of the critical value (similarly, p.a).(e) The keyboarding speed of secretaries at her company is comparable to the national average.
3. (a) m550(b) Nondirectional; the personnel director wants to know if there is a difference in either direction.(c) m 6¼ 50(d) z:055 61:96, z:015 62:58.
1. She would compare X with sample means expected through random sampling if the hypothesis m550 were true. If X is typical, she retains this hypothesis as a reasonable possibility; if X is very atypical, she rejects this hypothesis as unreasonable.
8. What sample values would be expected to occur in repeated random sampling and with what relative frequencies?9. (a) 5(b) .95
5. Treat the sample means obtained in Problem 4 as scores and calculate their standard deviation according to the procedures of Chapter 5.
3. Whether a sample is random depends on how it was selected, not on its composition.The chance factors involved in random sampling (sampling variation) occasionally lead to very atypical samples.
1. (a) the proverbial \person on the street" (i.e., people in general)(b) patrons at a local sports bar(c) No. Relative to the population, the sample undoubtedly overrepresents people who are interested in sports and frequent sports bars; other possible sources of bias are sex and age.
13. (a) a51:34, b5:0023, Y051:341:0023(X)(b) Val: 2.46, Mike: 2.97(c) .30(d) Val: 1.87 to 3.05, Mike: 2.38 to 3.56(e) z5(2:6522:46)=:305 þ :63; so proportion5:26(f) z5(2:0022:46)=:305 1:53; so proportion5:06(g) z5(2:5022:97)=:305 1:57; so proportion5:94
11. (a) Jean: 21.87, Albert: 12.26, Burrhus: 1.58(b) Jean: 21.33, Albert: 11.60, Burrhus: 1.41(c) Jean: 67:31( 1:33)(4:1)561:8, Albert: 67:31(1:60)(4:1)573:9, Burrhus: 67:31(:41)(4:1)569:0
7. (a) 8.83 (i.e., Y); given any value of X, the best prediction of Y is Ywhen r50.(b) Y0 58:8310(X)9. (a) 11.00(b) 2.33(c) 1.75(d) 2.20
5. Problem 3: For every inch increase in height at age 10, there is a corresponding increase of .94 inches in adult height. Problem 4: For every point increase in quiz 1 performance, there is a corresponding increase of .63 points in quiz 2 performance
4. (a) a52:84, b5 þ :63, Y0 52:841:63(X)(b) Keith: 7.9, Bill: 8.5, Charlie: 5.4, Brian: 4.1, Mick: 4.1; any discrepancies would be due to errors, rounding or otherwise.(c) Y0 56:00; the mean of the predicted Y scores will equal the mean of the actual Y scores.(d) S(Y 2Y0)50; the sum of the
3. (a) a521:941, b5 þ :94, Y0 521:941:94(X)(b) Jean: 61.89 inches, Albert: 73.92 inches, Burrhus: 69.03 inches.(c) Y0 for 10-year-old Jean is an estimate of the mean adult height for a large group of 10-year-olds, all of whom are 42.5 inches tall.
14. There is less variability in X and Y among experienced pilots than among cadets. Restriction of range reduces r.1. (a) Keith: 7.9, Bill: 8.5, Charlie: 5.4, Brian: 4.1, Mick: 4.1 (b) 12.1, 22.5, 12.6, 23.1 (c) 27.8 (d) It would be larger.
8. 1.28 12. (a) 2.60(b) no change(c) no change(d) r would be smaller(e) no change(f) no change(g) r would be smaller
7. (a) The numerator of the covariance would be smaller owing to the negative crossproduct for this pair of scores; consequently, the covariance would be smaller as well.(b) r also would be smaller.(c) We’d estimate that r would drop to 1.30 or so.
16. Mathematics achievement: If both distributions are normal, then approximately 59%(58.71%, to be precise) of the female scores fall below the mean for males. Verbal ability:Assuming normality, roughly 59%(59.10%) of the male scores fall below the female mean.
14. No. This will be true only in a symmetric distribution; in a skewed distribution, the mean (z50) and the median will not be the same. Where skew is positive, less than half the z scores in a distribution will be positive; where skew is negative, less than half the z scores will be negative. (If
13. By expressing all scores as z scores, you find the following order from best to worst: e,a,c, b, d.
15. (a) 9.10(b) 13.06(c) 1.22 and 2.23, respectively(d) In both cases, the difference between means is between one-fifth and one-quarter of a standard deviation. According to Cohen’s classification, these are \small" effect sizes.
(c) very high; his score falls between X 12S and X 13S.
(b) \average"; although above the mean, his score falls toward the center of the distribution.
14. (a) above the mean
(c) Samples may have the same mean but show different degrees of variability, or samples may have very different means but show the same degree of variability.
11. (a) (1) shows the least; (3) and (4) show the most.(b) for (1) X 58; S2 50; S50; for (2) X 58; S2 53:2; S51:79; for (3) X 58; S2 58;S52:83; for (4) X 51008; S2 58; S52:83
(b) will be slightly more variable. Distribution (c) will evidence considerable negative skew, and distribution (d) will have no variability whatsoever—everyone received a score of 50.
10. Distributions (a) and (b) both will be normal about a mean of 29, although distribution
14. The mean, because you must know each score to compute its value.
11. Those just below the median. If they improve sufficiently to pass the median, the median must increase to maintain an equal number of scores above and below.12. X 53964=50579:28; Mdn580:5; mode586
10. (a) First determine the original SX, which is equal to (n)(X)5(25)(23)5575. Reduce SX by nine points (i.e., 43 2 34 5 9) and divide by 25 to obtain the correct X 522:64.(b) The Mdn and mode would not be affected by this error. (Although technically the mode could change because of this error,
6. (a) negative skew(b) normal(c) bimodal(d) positive skew 7. (a) mode,Mdn,X(b) mode.Mdn.X(c) mode5Mdn5X(d) mode,Mdn,X
2. (a) mode58;Mdn510;X 512(b) no mode;Mdn515:5;X 517(c) mode511;Mdn511;X 511
12. Are the generalizations stated in Problem 11 limited to testing hypotheses about population correlation coefficients? (Use Table 19.1 to support your answer.)
(b) What generalization is illustrated by the comparison of Problems 10d and 10f?
11. (a) What generalization is illustrated by the comparison of Problems 10a and 10c?
*10. Determine the required sample size for each situation below:Effect Size (r) Desired Power Form of H1(a) .10 .80 Two-tailed(b) .60 .85 One-tailed(c) .70 .80 Two-tailed(d) .40 .99 One-tailed(e) .50 .75 Two-tailed(f) .40 .25 One-tailed
(e) Given the situation in Problem 9d and assuming a false H0, what proportion of such investigations—conducted repeatedly under identical conditions—would you expect to result in nonsignificance?Exercises 407
(d) Suppose you were able to recruit only 22 volunteers for your investigation. From Table 19.2, what would be your best estimate of the power of your test?
(c) Assuming a false H0, what proportion of such investigations—conducted repeatedly under identical conditions—would you nonetheless expect to result in nonsignificance?
(b) What is the required sample size to detect the specified effect at the desired level of power?
(a) What are your statistical hypotheses?
*9. You wish to correlate the number of errors committed on a problem-solving task with scores on a measure of impulsivity administered to a sample of college students. Your hypothesis is that students higher in impulsivity will tend to make more errors. You wish to detect an effect of r ¼ :40 and
(c) If you decided to adopt a ¼ :01, is the n in Problem 8a larger or smaller than it needs to be? (Explain.)
(b) If you were to use dependent samples, is the n in Problem 8a larger or smaller than it needs to be? (Explain.)
(a) What is the required sample size?
*8. You are planning an experiment. You set power at .85 and wish to detect an effect of at least d ¼ :30 (a ¼ :05, two-tailed).
7. The researcher in Problem 6, after examining his results, decides to increase his power by using a one-tailed test in the direction of the results. What is your response to this strategy?
6. A novice researcher is unable to recruit very many volunteers for his study. To increase his power, he decides to specify a larger effect size. What’s wrong with this approach?
(b) Imagine that the enrichment program, in truth, has an impact on the verbal intelligence of children. Given the estimate in Problem 5a, what proportion of such experiments—conducted repeatedly under identical conditions—would you expect to result in statistical significance?
(a) From Table 19.1, what would be your best estimate of the power of your test?
*5. In Problem 3, suppose you were interested in detecting a true difference of 9 IQ points but you ended up with only six children in each group.
(d) 21 IQ points?4. Repeat Problem 3 with power ¼ :50.
(c) 15 IQ points?
(b) 9 IQ points?power effect size factors affecting power significance versus importance sample size tables 406 Chapter 19 Statistical \Power" (and How to Increase It)
(a) 3 IQ points?
*3. You wish to determine the effects of a preschool enrichment program on verbal intelligence.Using a standardized instrument with m ¼ 100 and s ¼ 15, you intend to compare a group of children participating in the enrichment program with a matched group of nonparticipating children; a is set at
2. If the power of your test is .62 and you perform a particular experiment 50 times under identical circumstances, how many times would you expect to obtain statistically nonsignificant results?
(b) If, in truth, the effects of treatments A and B are identical, what is the power of the test? (Before responding, revisit the definition of power.)
(a) If the true value mA mB is 2.4, what is your best estimate of the power of the test?
*1. Consider a hypothetical situation in which an experiment to compare the effects of treatment A with those of treatment B is repeated 500 times under identical circumstances.A two-tailed test of H0: mA mB ¼ 0 is performed each time, and nonsignificant results are obtained 400 times.
18. In any 2 problem, what is the relationship between the row frequency and that row’s expected frequencies?

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