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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

(d) Conduct the F test (a ¼ :05) and present your results in an ANOVA summary table.
(c) What are the values for dfwithin, dfbetween, and dftotal?
(b) Compute SSwithin, SSbetween, and SStotal.
*12. Professor Loomis selects a sample of second-grade students from each of three schools offering different instructional programs in reading. He wishes to determine whether there are corresponding differences between these schools in the \phonological awareness"of their students. Professor
*11. Consider the assumptions underlying the F test for one-way analysis of variance (Section 16.12). Given the following data, do you believe the F test is defensible? (Explain.)X s n Group 1 75 21 36 Group 2 58 16 37 Group 3 60 10 18
10. Which case, Problem 7 or Problem 8, calls for the application of Tukey’s HSD test?(Explain.)9.344 Chapter 16 Comparing the Means of Three or More Independent Samples
9. (a) How many groups are there in Problem 7? (How do you know?)(b) What is the total sample size in Problem 7? (How do you know?)(c) How many groups are there in Problem 8? (How do you know?)(d) What is the total sample size in Problem 8? (How do you know?)
*8. Study the following ANOVA summary, and then provide the missing information for the cells designated a–f:Source SS df MS F p Between-groups 1104 (a) (b) 3.00 (c)Within-groups (d) (e) 184 Total 4416 (f)
7. Study the following ANOVA summary, and then provide the missing information for the cells designated a–f:Source SS df MS F p Between-groups (a) 3 (b) (c) (d)Within-groups 64 (e) (f)Total 349 19
*6. Determine F.05 and F.01 from Table C for each situation below:Total Sample Size Number of Groups(a) 82 3(b) 25 5(c) 120 4(d) 44 3
(b) Explain, within the context of this problem, what is meant by \inherent variation"and \differential treatment effects."
(a) Which is an estimate of inherent variation, and which is an estimate of differential treatment effects?
5. Consider s2 within and s2 between in Problem 4d.
*4. A researcher randomly assigns six students with behavioral problems to three treatment conditions (this, of course, would be far too few participants for practical study). At the end of three months, each student is rated on the ormality" of his or her behavior, as determined by classroom
(d) What’s wrong with expressing the alternative hypothesis as H1: m1 6¼ m2 6¼ m3 6¼ m4?
(c) List several possible ways in which H0 can be false.
(b) Why can’t H1 be expressed in symbolic form?
(a) Express H0 in symbolic form.
*3. You have designed an investigation involving the comparison of four groups.
2. List all the possible comparisons in Problem 1c (e.g., \1 vs. 2," \1 vs. 3," . . . ).
1. Using the formula k(k 1)=2, determine the number of t tests required to make all possible pairwise comparisons for each of the following conditions:(a) k ¼ 2(b) k ¼ 4(c) k ¼ 6(d) k ¼ 7
(b) What important principle, in addition to that illustrated in Problem 14, derives from this exercise?
(a) Compare the two sets of results with respect to sX1X2, the sample t ratio, and the appropriate statistical decision.
15. Recall the very low correlation between matched pairs in Problem 6 (r12 ¼þ:04). Reanalyze these data as if the scores were fromtwo independent groups of eight participants each.
14. The correlation calculated in Problem 3a (rpre;post ¼þ:68) indicates a considerable advantage to using the same participants under both conditions rather than two independent groups of five participants each. To see this advantage more directly, reanalyze the data in Problem 3 as if the
(c) Provide an alternative possible explanation (other than the experimental reading program itself) for the significantly better scores of the experimental children.
(b) Explain any important advantage(s) either design might have over the other.
(a) What is the essential difference between this research design and the design described in Problems 6 and 11?
*13. Parents of 14 entering first graders eagerly volunteer their children for the tryout of a new experimental reading program announced at a PTA meeting. To obtain an \equivalent"group for comparison purposes, each experimental child is matched with a child in the regular program on the basis of
12. You wish to see whether students perform differently on essay tests and on multiplechoice tests. You select a sample of eight students enrolled in an introductory biology course and have each student take an essay test and a multiple-choice test. Both tests cover the same unit of instruction
11. An exercise physiologist compares two cardiovascular fitness programs. Ten matched pairs of out-of-shape adult volunteers are formed on the basis of a variety of factors such as sex, age, weight, blood pressure, exercise, and eating habits. In each pair, one individual is randomly assigned to
10. A psychological testing firm wishes to determine whether college applicants can improve their college aptitude test scores by taking the test twice. To investigate this question, a sample of 40 high school juniors takes the test on two occasions, three weeks apart. The following are the
*9. Is one Internet search engine more efficient than another? You ask each of seven student volunteers to find information on a specific topic using one search engine (search 1) and Training Program 1: X1 ¼ 56:3 SS1 ¼ 538 Training Program 2: X2 ¼ 44:3 SS2 ¼ 354 r12 ¼ þ:04 Exercises 315 then
*8. Consider Problem 5:(a) Without performing any calculations, what one value do you know for certain would not fall in a 95% confidence interval for m1 m2? (Explain.)(b) Construct and interpret a 95% confidence interval for m1 m2.
7. Consider Problem 6:(a) Without performing any calculations, what one value do you know for certain would fall in a 99% confidence interval for m1 m2? (Explain.)(b) Construct and interpret a 99% confidence interval for m1 m2.(c) What two factors contribute to the width of the confidence
*6. The sales manager of a large educational software company compares two training programs offered by competing firms. She forms eight matched pairs of sales trainees on the basis of their verbal aptitude scores obtained at the time of initial employment; she randomly assigns one member of each
*5. Professor Civiello wishes to investigate problem-solving skills under two conditions: solving a problem with and without background music. In a carefully controlled experiment involving six research participants, Dr. Civiello records the time it takes each participant to solve a problem when
4. Repeat Problem 3c, except use the direct-difference method.(a) What are the statistical hypotheses?(b) Compute D, SSD, and sD.(c) Test H0.(d) Draw final conclusions.(e) Give the symbols for the quantities from Problem 3 that correspond to mD, D, and sD.(f) Compare your results to those for
*3. The following are scores for five participants in an investigation having a pretest–posttest design:Participant A B C D E Pretest 12 6 8 5 9 Posttest 9 8 6 1 6(a) Compute SSpre, SSpost, and rpre,post.(b) From SSpre and SSpost, determine s2 pre and s2 post.(c) Compute sXpreXpost(d) Test H0:
2.(a) How can the use of matched pairs be of help statistically?(b) What one single value can you compute from the results of a matched-pairs investigation that will tell you the degree to which the matching has helped?
*1. Suppose you wish to use high school seniors for an investigation concerning the relative efficacy of two treatment conditions for reducing test anxiety. You draw a random sample of seniors from a local high school, randomly assign subjects to treatments, and then conduct the statistical test.
(b) Describe some of the limits to any statistical inferences based on the study.
(a) Is the statement a statistical or nonstatistical inference? (Explain.)
21. Suppose the following statement were made on the basis of the significant difference reported in Problem 13: \Statistics show that women are higher in emotional intelligence than men."
20. Is randomization the same as random sampling? (Explain.)
(b) What are some other possible explanations—other than whether a child attended preschool—for a significant difference in sociability in Problem 9?*19. Examine Problems 8, 9, 10, 14, and 16. In which would it be easiest to clarify causal relationships? (Explain.)
(a) For which investigation would it be easier to clarify the relationship between cause and effect? (Explain.)
18. Compare the investigation described in Problem 9 with that in Problem 14. Suppose a significant difference had been found in both—in favor of the children who attended preschool in Problem 9 and in favor of Group 2 in Problem 14.
(b) Calculate and interpret the effect size, oˆ 2.
17. From the data given in Problem 16:(a) Compute and interpret the effect size, d; evaluate its magnitude in terms of Cohen’s criteria and in terms of the normal curve.
16. The director of Academic Support Services wants to test the efficacy of a possible intervention for undergraduate students who are placed on academic probation. She randomly assigns 28 such students to two groups. During the first week of the semester, students in Group 1 receive daily
(d) Now construct a 99% confidence interval for m1 m2, given the data in Problem 14.Any surprises?
(c) Without performing any calculations, comment on whether a 99%confidence interval estimated from the same data would include zero.
*15. (a) Suppose you constructed a 95% confidence interval for m1 m2, given the data in Problem 14. What one value do you already know will reside in that interval?(Explain.)(b) Now construct a 95% confidence interval for m1 m2, given the data in Problem 14.Any surprises?
*14. A high school social studies teacher decides to conduct action research in her classroom by investigating the effects of immediate testing on memory. She randomly divides her class into two groups. Group 1 studies a short essay for 20 minutes, whereas Group 2 studies the essay for 20 minutes
*13. You read the following in a popular magazine: \A group of college women scored significantly higher, on average, than a group of college men on a test of emotional intelligence."(Limit your answers to statistical matters covered in this chapter.)(a) How is the statistically unsophisticated
12. Parametric statistical tests are tests that are based on one or more assumptions about the nature of the populations from which the samples are selected. What assumptions are required in the t test of H0: m1 m2 ¼ 0?
11. From the data given in Problem 10:(a) Compute and interpret the effect size, d; evaluate its magnitude in terms of Cohen’s criteria and in terms of the normal curve.(b) Calculate and interpret the effect size, oˆ 2.
*10. You are investigating the possible differences between eighth-grade boys and girls regarding their perceptions of the usefulness and relevance of science for the roles they see themselves assuming as adults. Your research hypothesis is that boys hold more positive perceptions in this regard.
9. An educational psychologist is interested in knowing whether the experience of attending preschool is related to subsequent sociability. She identifies two groups of first graders:those who had attended preschool and those who had not. Then each child is assigned a sociability score on the basis
8. Does familiarity with an assessment increase test scores? You hypothesize that it does.You identify 11 fifth-grade students to take a writing assessment that they had not experienced before. Six of these students are selected at random and, before taking the assessment, are provided with a
*7. For each of the following cases, give the critical value(s) of t:(a) H1: m1 m2 > 0, n1 ¼ 6, n2 ¼ 12, a ¼ :05(b) H1: m1 m2 6¼ 0, n1 ¼ 12, n2 ¼ 14, a ¼ :01(c) H1: m1 m2 < 0, n1 ¼ 14, n2 ¼ 16, a ¼ :05(d) H1: m1 m2 6¼ 0, n1 ¼ 19, n2 ¼ 18, a ¼ :01
*6. From the data given in Problem 5:(a) Compute and interpret the effect size, d; evaluate its magnitude in terms of Cohen’s criteria and in terms of the normal curve.(b) Calculate and interpret the effect size, oˆ 2:
*5. The following results are for two samples, one from Population 1 and the other from Population 2:from Population 1 : 3; 5; 7; 5 from Population 2 : 8; 9; 6; 5; 12(a) Compute SS1 and SS2.(b) Using the results from Problem 5a, compute the pooled variance estimate.(c) Using the result from Problem
4. Assume H0: m1 m2 ¼ 0 is true. What are the three defining characteristics of the sampling distribution of differences between means?
(b) Describe how the results used to construct the sampling distribution could be used to obtain an estimate of sXAXB.
(a) Describe how, through a series of sampling experiments, you could construct a fairly accurate picture of the sampling distribution of XA XB for samples of size nA ¼ 5 and nB ¼ 5.
*3. Consider two large populations of observations, A and B. Suppose you have unlimited time and resources.
2. A graduate student wishes to compare the high school grade-point averages (GPAs) of males and females. He identifies 50 brother/sister pairs, obtains the GPA for each individual, and proceeds to test H0: mmales mfemales ¼ 0. Are the methods discussed in this chapter appropriate for such a
*1. Translate each of the following into words, and then express each in symbols in terms of a difference between means relative to zero:(a) mA > mB(b) mA < mB(c) mA ¼ mB(d) mA 6¼ mB
2. Repeat the process above, but instead of GPA as the dependent variable, use performance on the reading and mathematics exams.
1. Access the students data set, which contains grade-point averages (GPA) and television viewing information (TVHRSWK) for a random sample of 75 tenth-grade students. Test whether there is a statistically significant difference in GPA between students who watch less than two hours of television
25. How do you explain the considerable width of the resulting confidence intervals in Problem 24?
*24. From the data in Problems 8a and 8b, determine and interpret the respective 95%confidence intervals for m.
23. Suppose the director in Problem 22 is criticized for conducting a t test in which there is evidence of nonnormality in the population.(a) How do these sample results suggest population nonnormality?(b) What is your response to this critic?
*22. Fifteen years ago, a complete survey of all undergraduate students at a large university indicated that the average student smoked X ¼ 8:3 cigarettes per day. The director of the student health center wishes to determine whether the incidence of cigarette smoking at his university has
21. The expression \p < :001" occurs in the results section of a journal article. Does this indicate that the investigator used the very conservative level of significance a ¼ :001 to test the null hypothesis? (Explain.)
*20. Suppose a ¼ 0:5 and the researcher reports that the sample mean \approached significance."(a) What do you think is meant by this expression?(b) Translate the researcher’s statement into symbolic form involving a p value.Exercises 273
19. Translate each of the following statements into symbolic form involving a p value:(a) \The results did not reach significance at the .05 level."(b) \The sample mean fell significantly below 50 at the .01 level."(c) \The results were significant at the .001 level."(d) \The difference between the
18. Repeat Problem 17, this time assuming that the investigator has in mind a ¼ :01.
*17. For each of the following sample t ratios, report the p value relative to a suitable \landmark"(as discussed in Section 13.8). Select among the landmarks .10, .05, and .01, and assume that the investigator in each case has in mind a ¼ :05.(a) H1: m < 100, n ¼ 8, t ¼ 2:01(b) H1: m 6¼ 60, n
(b) If any other sample were drawn, what should be done differently so that a t test would be appropriate?
(a) From the sample results, what makes you think that the proposed t test may be inappropriate?
*16. The following are the times (in seconds) that a sample of five 8-year-olds took to complete a particular item on a spatial reasoning test: X ¼ 12:3 and s ¼ 9:8. The investigator wishes to use these results in performing a t test of H0: m ¼ 8.
15. Using the data in Problem 8b, an investigator tests H0: m ¼ 11:25 against H1: m 6¼ 11:25.(a) Determine t.01.(b) Perform the statistical test.(c) Draw final conclusions.
*14. Consider the data in Problem 8a. Suppose the researcher wants to test the hypothesis that the population mean is equal to 72; she is interested in sample departures from this mean in either direction.(a) Set up H0 and H1.(b) Determine t.05.(c) Perform the statistical test.(d) Draw final
13. The task in a particular concept-formation experiment is to discover, through trial and error, the correct sequence in which to press a row of buttons. It is determined from the nature of the task that the average score obtained by random guessing alone would be 20 correct out of a standard
*12. For each of the following instances, locate the regions of rejection and the sample results on a rough distribution sketch; perform the test; and give final conclusions about the value of m.(a) H0: m ¼ 10, H1: m 6¼ 10, a ¼ :10, sample: 15, 13, 12, 8, 15, 12(b) Same as Problem 12a except a
11. From Table B and for df ¼ 25, find the proportion of t values that would be:(a) less than t ¼ 1:316(b) less than t ¼ þ1:316(c) between t ¼ 2:060 and t ¼ þ2:060(d) between t ¼ 1:708 and t ¼ þ2:060
*10. From Table B, identify the centrally located limits, for df ¼ 8, that would include:(a) 90% of t values(b) 95% of t values(c) 99% of t values
9. From Table B, identify the value of t that for df ¼ 15:(a) is so high that only 1% of the t values would be higher(b) is so low that only 10% of the t values would be lower
*8. Compute the best estimate of s and sX for each of the following samples:(a) percentage correct on a multiple-choice exam: 72, 86, 75, 66, 90(b) number of points on a performance assessment: 2, 7, 8, 6, 6, 11, 3
7. Comment on the following statement: For small samples selected from a normal population, the sampling distribution of means follows Student’s t distribution.
*6. Suppose that df ¼ 3. How do the tails of the corresponding t distribution compare with the tails of the normal curve? Support your answer by referring to Tables A and B in Appendix C (assume a ¼ :10, two-tailed).
5. Why is the t distribution a whole family rather than a single distribution?
Use an a of .05 for both analyses.estimated standard deviation estimated standard error of the mean t ratio Student’s t distribution degrees of freedom family of distributions critical values of t one-sample t test t distribution versus z distribution normality assumption\landmark" p values
(b) Determine whether mean per-pupil expenditure is significantly different from $5500.
(a) Determine whether mean teacher salary is significantly different from $32,000.
2. Access the fiscal data file, which contains average teacher salaries and per-pupil expenditures from 60 school districts in a southeastern state. Use the one-sample t test to conduct the following tasks.
1. Access gosset, a data set used by W. S. Gosset circa 1908. The file contains one variable, ADDHRS, which represents the additional hours of sleep gained by 10 patients after exposure to laevohysocyamine hydrobromide. Using the one-sample t test, determine whether the experimental treatment

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