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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
*17. Consider the data given in Problem 16. Test the null hypothesis that all candidates are equally popular (a ¼ :05).
(d) What is your general conclusion from this 2?
(c) Compute 2 (using Formula [18.1]) and complete the test at a ¼ :05.
(b) Determine fo and fe for each of the 12 cells of this 3 4 contingency table. Present this information in a 3 4 table that includes row totals, column totals, and the grand total. (For each row, does Sfe ¼ frow?)
(a) Stated very generally, what is the null hypothesis of independence in this situation?
*16. In a particular county, a random sample of 225 adults are asked for their preferences among three individuals who wish to be the state’s next commissioner of education.Respondents also are asked to report their annual household income. The results:Candidate Household Income Jadallah Yung
15. Using the data given in Problem 14, calculate 2 from Formula (18.7).
(f) Translate each obtained frequency into a proportion based on its row frequency.What interpretation seems likely?
(e) What is your general conclusion from this significant 2?
(d) Compute 2 (using Formula [18.1]) and complete the test at a ¼ :05.
(c) Determine fo and fe for each of the four cells of this 2 2 contingency table. Present this information in a 2 2 table that includes row totals, column totals, and the grand total.
(b) State H1 (nondirectional) in words for each H0 in Problem 14a.
(a) Given this situation, state two equivalent expressions (in words) for the null hypothesis of independence in terms of proportions.
14. Is sexual activity among adolescent females related to whether one is a smoker or nonsmoker?Harriet Imrey, in an article appearing in The Journal of Irreproducible Results(Imrey, 1983), provided the following data from a sample of 508 girls between the ages of 14 and 17:Sexually Active Sexually
*13. Forty volunteers participate in an experiment on attitude change. An attitude item is completed by these individuals both before and after they watch a spirited debate on the topic. The following data are obtained:390 Chapter 18 Making Inferences From Frequency Data Response to Attitude
11. (a) Why is a directional H1 possible in the Problem 10 scenario?(b) Offer an example of a directional H1 (in words).*12. Using the data given in Problem 10, calculate 2 from Formula (18.7).
(e) What is your general interpretation of this finding, based on a comparison of the fo’s and fe’s?
(d) What is your general conclusion from this 2?
(c) Compute 2 (using Formula [18.1]) and complete the test at a ¼ :05 and at a ¼ :01 (two-tailed).
(b) Determine fo and fe for each of the four cells of this 2 2 contingency table. Present this information in a 2 2 table that includes row totals, column totals, and the grand total. (For each row, does Sfe ¼ frow?)
(a) Given this situation, state (in words) the null hypothesis of independence in terms of proportions.
*10. A sample of 163 prospective voters is identified from both rural and urban communities.Each voter is asked for his or her position on the upcoming \gay rights" state referendum.The results are as follows:In favor Opposed Rural 35 55 Urban 53 20
9. Give the critical 2 values for testing the null hypothesis of independence at a ¼ :05 and a ¼ :01 for each of the following contingency tables:(a) 2 3 table(b) 2 6 table(c) 3 5 table(d) 2 2 table
(d) Do these results prove the die is loaded and thus unfair? (Explain.)
(c) Compute 2, complete the test (a ¼ :05), and draw your conclusion regarding this die.
8. You wish to determine whether a friend’s die is \loaded." You roll the die 120 times and obtain the following results:Side coming up: 1 2 3 4 5 6 Number observed: 16 16 10 20 28 30(a) Give H0 for this situation (use fractions).(b) Can a single H1 be written? (Explain.)
(c) Draw final conclusions.Exercises 389
(b) Compute 2 and complete the test at a ¼ :01.
7. The 72 college students in an educational psychology class take a multiple-choice midterm exam. The professor wishes to test the hypothesis that students guessed at random on the options for question 36. The frequency of responses for that item was as follows:Option: A B C D Frequency: 15 40 5
(b) Use Formulas (18.3) and (18.4) to construct and interpret a 95% confidence interval for p.(c) Use Figure 18.3 to obtain an approximate confidence interval for p. How does this compare to what you obtained in Problem 6b?
(a) Calculate the proportion of gang members who dropped out of high school.
*5. Suppose it is known in a large urban school district that four out of five teenagers who join a gang subsequently drop out of high school. A \stay in school" intervention is instituted for a sample of 45 gang members. It is later found that 30 of these students have remained in school and 15
4. In the 2 test, why is it that only the area in the upper tail of the 2 distribution is of interest?
(e) From this 2, what is your general conclusion? That is, do the four achievement tests appear to differ in popularity?
(d) Compute 2 and test H0 at a ¼ :01.
(c) Compute the expected frequencies under H0. (Do they sum to n?)
(b) Can H1 be written in a single symbolic statement? (Explain.)
(a) Give, in symbolic form, two equivalent statements of H0 for this situation.
*3. A researcher wishes to determine whether four commercially available standardized achievement tests differ in their popularity. He obtains a random sample of 60 school districts in his region of the country and asks each superintendent which standardized achievement test is used. (Assume that
2. (a) For which H0 in Problem 1 would a directional H1 be possible? (Explain.)(b) What is the one-tailed value for 2:05 and for 2:01?
(b) compute the necessary tests at a ¼ :05;(c) draw final conclusions.388 Chapter 18 Making Inferences From Frequency Data
(a) construct a contingency table that includes both observed and expected frequency counts;
3. Examine whether there is a relationship between gender and eighth-grade math course selection.In doing so,
2. Repeat the test above, this time testing the observed proportions against .33 algebra and .67 general math.
1. Use this sample of students to test whether eighth graders are equally likely to take either algebra or general math (i.e., .50 take algebra, .50 take general math).(a) provide H0;(b) compute x2;(c) complete the test at a ¼ :05.
*1. Give the critical 2 values and df for testing each of the following null hypotheses for one-variable problems at a ¼ :05 and a ¼ :01.(a) H0 : p1 ¼ p2 ¼ p3 ¼ p4(b) H0 : p1 ¼ :10, p2 ¼ :10, p3 ¼ :80(c) H0 : p1 ¼ :25, p2 ¼ :75(d) H0 : p1 ¼ p2(e) H0 : p1 ¼ :50, p2 ¼ p3 ¼ p4 ¼ p5 ¼
(b) compute the necessary tests at a ¼ :05;(c) draw final conclusions
(a) construct a contingency table that includes both observed and expected frequency counts;
3. Examine whether there is a relationship between gender and eighth-grade math course selection.In doing so,
2. Repeat the test above, this time testing the observed proportions against .33 algebra and .67 general math.
1. Use this sample of students to test whether eighth graders are equally likely to take either algebra or general math (i.e., .50 take algebra, .50 take general math).(a) provide H0;(b) compute x2;(c) complete the test at a ¼ :05.
*17. A researcher believes that the ability to identify constellations of stars in the night sky is related to spatial reasoning ability. She obtains the correlation between scores on a spatial reasoning test (X) and the number of constellations correctly identified (Y). She calculates this
16. Using a sample of 120 high schools, a researcher obtains a correlation of r ¼:52 (p < :05)between average teacher salary (X) and the proportion of students who drop out (Y).Considering the earlier discussion of correlation and causation (Section 7.6), what do you believe is the most likely
(b) What would be the statistical consequence of computing Pearson r from these data? (No calculations necessary.)13.*10.Exercises 363
(a) Would the Pearson r be an appropriate measure of association for these data?(Explain.) (Hint: Construct a scatterplot.)
*15. For a sample of her 10 students, an instructor correlates \test anxiety" (X) with \percent correct" (Y) on the recent midterm. The data are as follows:Student A B C D E F G H I J Percent correct 73 92 55 84 64 88 69 96 59 77 Test anxiety 35 26 48 21 10 30 42 25 4 16
*14. Why is the sample r alone sufficient for adequate interpretation when the sample size is quite large (say over 300 or 400 cases), whereas an interval estimate is recommended for smaller samples?
(b) Suppose the correlation between two variables is reported as \significant" for a sample of 1000 cases. Is it possible, without knowing the actual value of r, to make an adequate interpretation concerning the true degree of relationship from this information alone? (Explain.)
13. (a) Suppose the correlation between two variables is reported as ot significant"for a sample of 1000 cases. Is it possible, without knowing the actual value of r, to make an adequate interpretation concerning the true degree of relationship from this information alone? (Explain.)
12. Consider the confidence intervals you estimated in Problems 9 and 11. If in each of those cases you instead had tested H0: r ¼ 0 (a ¼ :05, two-tailed), which sample correlation coefficients would have resulted in nonsignificance? (Explain.) (Note: Answer this question simply by examining the
11. Use Figure 17.7 to determine (as accurately as you can) the 95% confidence interval for r in each of the following instances:(a) r ¼þ:35, n ¼ 50(b) r ¼ :45, n ¼ 15(c) r ¼þ:78, n ¼ 10(d) r ¼ :52, n ¼ 100
(b) Now compare the widths of the intervals obtained in Problems 9c–9e. What is the corresponding generalization concerning the sampling variation of the correlation coefficient?
*10. (a) Compare the widths of the intervals obtained in Problems 9a–9c. What generalization concerning the sampling variation of the correlation coefficient is suggested by this comparison?
*9. Use Figure 17.7 to determine (as accurately as you can) the 95% confidence interval for r in each of the following instances:(a) r ¼þ:90, n ¼ 10(b) r ¼þ:50, n ¼ 10(c) r ¼þ:20, n ¼ 10 362 Chapter 17 Inferences About the Pearson Correlation Coefficient(d) r ¼þ:20, n ¼ 30(e) r ¼þ:20,
(c) What particular weakness of the personnel director’s study is illustrated by your answer to Problem 8b?
(b) Use an interval estimation approach to evaluate the sample results (95% level of confidence).
(a) Use a significance testing approach (Table E) to evaluate the sample result (a ¼ :05).
*8. An education professor has 15 college seniors who are doing their student teaching.They also recently took a teacher certification test required by the state. The professor obtains a correlation of +.40 between these test scores and ratings of student-teaching performance that were provided by
(b) What does the statement really mean? (Be precise; use appropriate symbols and statistical terminology.)(c) What single piece of additional information would be most necessary for adequately interpreting the result claimed?
(a) How might such a statement be misinterpreted by the statistically unsophisticated?
7. You read in a review article: \A researcher found a significant positive correlation between popularity and IQ for a large sample of college students."
6. For each of the following cases, give the size of the sample r required for statistical significance:(a) n ¼ 5, a ¼ :05, one-tailed test(b) n ¼ 24, a ¼ :05, two-tailed test(c) n ¼ 42, a ¼ :01, two-tailed test(d) n ¼ 125, a ¼ :05, two-tailed test(e) n ¼ 1500, a ¼ :05, one-tailed test(f)
5. Suppose that the researcher in Problem 4, while presenting her results at a conference, said the following: \Interestingly, the obtained correlation was negative. That is, there is a slight tendency for children of higher socioeconomic backgrounds to be lower in selfesteem."What would be your
4. Using a sample of 26 twelve-year-olds from diverse backgrounds, a researcher conducts an exploratory study of the relationship between self-esteem and socioeconomic status.She obtains a sample correlation of r ¼ :12.(a) Specify the statistical hypotheses.(b) Specify the critical r value and
*3. For the five situations in Problem 2, provide the critical r value and the statistical decision regarding H0: r ¼ 0. (Do the statistical decisions agree across the two problems?)
(b) Repeat 1a, but with respect to the relationship between science self-efficacy and performance on the state exam.linear association sampling distribution of r standard error of r normal bivariate distribution heteroscedasticity critical values of r coefficient of determination Exercises 361
(a) Compute the Pearson r between classroom support and performance on the district exam, and determine its statistical significance(a ¼ :05). (Decide whether you wish to conduct a one-tailed or a two-tailed test.) If significant, interpret the magnitude of r in terms of the coefficient of
1. Access the ch17 data file, which contains information on a nonrandom sample (n ¼ 64) of eighthgrade students from a rural school district in the Midwest. The data include science test scores from state and district assessments, as well as two selfreport measures: science self-efficacy and
*2. For each situation below, provide the following: sr, sample t ratio, critical t value, and the statistical decision regarding H0: r ¼ 0. (Assume that the obtained correlation is in the direction of H1.)(a) r ¼ :38, n ¼ 30, a ¼ :05, two-tailed test(b) r ¼þ:60, n ¼ 10, a ¼ :05,
*1. Suppose that a friend wishes to test H0: r ¼ :25 and asks for your assistance in using the procedures described in this chapter. What would be your response?
(b) What are possible explanations—other than instructional program—for the significant F ratio in Problem 12?
(a) For which investigation is it more difficult to argue a cause-and-effect relationship?(Explain.)
*20. Compare the investigation described in Problem 12 with that in Problem 18.
*19. (a) Construct a 95% confidence interval for each mean difference in Problem 18.(b) How do the obtained confidence intervals compare with your decisions regarding the null hypotheses in Problem 18?
*18. You wish to compare the effectiveness of four methods for teaching metacognitive strategies to elementary school children. A group of 40 fifth graders is randomly divided into four subgroups, each of which is taught according to one of the different methods. You then individually engage each
(c) Apply the Tukey test (a ¼ :05) to all possible pairwise comparisons between means and draw final conclusions.(d) Repeat Problem 17c using a ¼ :01.
(b) Display the differences between the means for all possible pairs of samples as illustrated in Section 16.8 (step 3).
(a) Use the formula k(k 1)=2 to determine the number of all possible pairs of means.
*17. A one-way ANOVA is carried out using the performance scores from five different treatment groups of nine cases each. A significant F is obtained. For this analysis s2 within ¼ 20:5, and the treatment group means are as follows:X1 ¼ 20:3; X2 ¼ 12:2; X3 ¼ 15:3; X4 ¼ 13:6; and X5 ¼ 19:1
(b) Complete the F test (a ¼ :01), state your statistical decision regarding H0, and present the results in a summary table.14.13.Exercises 345
16. A study is performed using observations from five samples of 20 cases each. The following are partial results from a one-way analysis of variance: SSbetween ¼ 717 and SStotal ¼ 6861.(a) Compute s2 within and s2 between.
15. Suppose you obtained a significant F ratio and now wish to apply the Tukey test. However, you have unequal n’s : n1 ¼ 15, n2 ¼ 18, and n3 ¼ 14. Compute the harmonic mean.
(c) Interpret the confidence interval for m2 m3.
(b) How do these confidence intervals compare with the answers to Problem 13?
14. (a) Construct a 95% confidence interval for each of the mean differences in Problem 12.
(b) State your conclusions.
13. (a) Apply Tukey’s HSD test (a ¼ :05) to the results of Problem 12.
(g) What is your substantive conclusion from this analysis? (Is your analysis complete?)
(f) Compute and interpret oˆ 2 from these data.
(e) What is your statistical decision regarding H0?
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