1. How will an increase in the sample size affect the width of the confidence interval?

Transcription

1 Study Guide Concept Questions 1. How will an increase in the sample size affect the width of the confidence interval? 2. How will an increase in the sample size affect the power of a statistical test? 3. How will an increase in the sample size affect the p-value? 4. When planning a future study that will report a confidence interval result, how does decreasing the desired confidence interval width affect the sample size requirement? 5. When planning a future study that will report a confidence interval result, how does increasing the desired level of confidence affect the sample size requirement? 6. When planning a future study that will report a hypothesis testing result, how does increasing the desired power affect the sample size requirement? 7. When planning a future study that will report a hypothesis testing result, how does decreasing the alpha level affect the sample size requirement? 8. When planning a future study that will report a hypothesis testing result, how does decreasing the value of the expected effect size affect the sample size requirement? k 9. When planning a future study to estimate μ, μ 1 μ 2, or j=1 c j μ j, how does using a larger value of σ 2 affect the sample size requirement? 10. When planning a future study to test H0: μ = h, H0: μ 1 = μ 2, or H0: j=1 c j μ j = 0, how does using a larger value of σ 2 affect the sample size requirement? 11. When planning a future study to estimate δ, how does using a larger value of δ 2 affect the sample size requirement? 12. Why are narrower confidence intervals preferred over wider confidence intervals? k

2 13. Why would a 95% confidence interval be preferable to a 90% confidence interval? 14. Why is higher power desirable? 15. What are some ways to obtain a planning value for σ? 16. How do you modify the sample size formulas for testing or estimating a difference in means or a difference in proportions when planning a future study that will report v pairwise tests or confidence intervals? 17. How does the planning value of the correlation between the measurements in a paired-samples designs affect the sample size requirement for a confidence interval for μ 1 μ 2 or a test of H0: μ 1 = μ 2? 18. How does the planning value of ρ yx affect the sample size requirement for estimating ρ yx with desired precision or testing a hypothesis regarding the value of ρ yx with desired power? 19. How does the planning value of π affect the sample size requirement for estimating π with desired precision or testing H0: π = h with desired power? 20. What planning value of π will give the largest sample size requirement? 21. How does the number of control variables affect the sample size requirement for a confidence interval or hypothesis test regarding the value of a partial correlation? 22. Why should researchers avoid using unnecessarily large samples? 23. What are some consequences of using a sample size that is too small? 24. What are the sample size implications of sampling from a diverse study population rather than a more homogeneous study population? 25. What are the advantages of using equal sample sizes in a multiple group design? 26. Why are unequal sample sizes in a multiple group design sometimes justified? 27. How does the range of x values affect the sample size requirement for testing or estimating a slope in a fixed-x model?

3 28. When is a two-sage sample size analysis useful? 29. What is the effect of using a larger assurance probability on the sample size requirement? 30. What is the advantage of using an upper confidence limit for a population variance rather than a sample variance as a variance planning value? Computation Problems 1. How large of a sample is needed to obtain a 95% confidence interval for μ with a width of 5.0 based on a variance planning value of 38? 2. A researcher plans to test H0: μ = 200 at α =.05 using a sample size of n = 30. What is the power of the test if μ = 240 and σ = 60? 3. What is the expected width of a 90% confidence interval for μ in a sample size of 100 and a variance planning value of 36? 4. What sample size is required to test H0: μ = 50 with power =.90, σ 2 = 40, α =.05, and μ = 45? 5. For a 2-group design, what is the sample size requirement per group to obtain a 95% confidence interval for μ 1 μ 2 with a width of 8.0, and a variance planning value of 50? 6. For a 2-group design, what sample size is required per group to test H0: μ 1 = μ 2 with power =.80, σ 2 = 100, α =.05, and μ 1 μ 2 = 7? 7. A researcher plans to test H0: μ 1 = μ 2 at α =.05 using a 2-group design and sample sizes of n 1 = 10 and n 2 = 20. What is the power of the test if μ 1 μ 2 = 5 and σ 1 = σ 2 = 15? 8. A researcher plans to compute a 95% confidence interval for μ 1 μ 2 using a 2-group design and sample sizes of n 1 = 30 and n 2 = 30. What is expected confidence interval width with σ = 8? 9. For a 2-group design, what is the sample size requirement per group to obtain a 95% confidence interval for δ with a width of 0.5 and δ = 0.8?

4 10. For a between-subjects design, what is the sample size requirement per group to obtain a 95% confidence interval for (μ 1 + μ 2 )/2 μ 3 with a width of 3.0 and a variance planning value of 10? 11. A researcher plans to test H0: μ 1 μ 2 μ 3 + μ 4 at α =.01 using a between-subjects design and 12 participants per group. What is the power of the test if μ 1 μ 2 μ 3 + μ 4 = 8 and all within-group standard deviations are assumed to be 15? 12. A researcher plans to compute a 99% confidence interval for (μ 1 + μ 2 )/2 (μ 3 + μ 4 )/2. What is the expected confidence interval width for σ = 50? 13. For a between-subjects design, what sample size per group is required to test H0: (μ 1 + μ 2 )/2 (μ 3 + μ 4 )/2 with (μ 1 + μ 2)/2 (μ 3 + μ 4)/2 = 4, power =.90, σ 2 = 50, and α = For a between-subjects design, what is the sample size requirement per group to obtain a 95% confidence interval for a standardized contrast of (μ 1 + μ 2 )/2 μ 3 with a width of 0.6 and φ = 1.0? 15. Suppose a researcher obtained a 95% confidence interval for φ in a between-subjects design using 50 participants per group in a first-stage sample and obtained a confidence interval width of 1.3. How many participants should be sampled per group in the second stage to reduce the 95% confidence interval width to 0.8? 16. For a 2-level with-subjects design, what is the sample size requirement to obtain a 95% confidence interval for μ 1 μ 2 with a width of 1.0, a variance planning value of 3, and a correlation planning value of.75? 17. For a 2-level with-subjects design, what is the sample size requirement to obtain a 95% confidence interval for δ with a width of 0.4, δ = 1.5, and ρ 12 =.80? 18. A researcher plans to compute a 95% confidence interval for μ 1 μ 2 using a withinsubjects design and a sample size of 25. What is the expected confidence interval width with σ 2 = 15 and ρ 12 =.75? 19. For a 2-level with-subjects design, what sample size is required to test H0: μ 1 = μ 2 with power =.85, σ 2 = 180, α =.05, ρ 12 =.80, and μ 1 μ 2 = 5? 20. What sample size is required to test H0: π =.5 with power =.95, α =.05, and π =.75?

5 21. What sample size is required to obtain a 95% confidence interval for π with a width of.2 and π =.6? 22. A researcher is planning to test H0: π =.25 at α =.05 in a sample of n = 50. What is the expected power of this test for π =.45? 23. A researcher is planning to compute a 95% confidence interval for π in a sample of n = 50. What is the expected confidence interval width for π =.75? 24. For a 2-group design, what is the sample size requirement per group to obtain a 95% confidence for π 1 π 2 with a width of 0.3 using π 1 =.3 and π 1 =.5? 25. For a 2-group design, what sample size is required per group to test H0: π 1 = π 2 with power =.90, α =.05, for π 1 =.6 and π 1 =.75? 26. A researcher plans to test H0: π 1 = π 2 at α =.05 using a 2-group design and sample sizes of n 1 = 50 and n 2 = 50. What is the power of the test if π 1 =.25 and π 2 =.4? 27. For a 2-level within-subject design, what is the sample size requirement to test H0: π 1 = π 2 at α =.05 with power of.8 using π 1 =.1, π 1 =.2, and ρ =.6? 28. For a 2-level within-subject design, what is the sample size requirement to obtain a 95% confidence for π 1 π 2 with a width of 0.25 using π 1 =.3, π 1 =.5, and ρ =.5? 29. For a between-subjects design, what is the sample size requirement per group to obtain a 95% confidence interval for (π 1 + π 2 )/2 (π 3 + π 4 + π 5 )/3 with a width of 0.2 and planning values for π 1, π 2, π 3, π 4, and π 5 equal to.2,.2,.4,.4, and.4, respectively. 30. What sample size is required to test H0: ρ yx =.4 with power =.90, α =.05, and ρ yx =.6? 31. How large of a sample is needed to obtain a 95% confidence interval for ρ yx with a width of.2 using ρ yx =.5? 32. A researcher is planning to test H0: ρ yx = 0 using a sample size of 50. What is the expected power of this test at α =.05 and ρ yx =.2? 33. A researcher plans to compute a 95% confidence interval for ρ yx in a sample of n = 100. What is the expected confidence interval width with ρ yx =.4?

6 34. How large of a sample is needed to obtain a 95% confidence interval for squared multiple correlation for 3 predictor variables with a desired confidence interval width of.2 and a squared multiple correlation planning value of.25? 35. How large of a sample is needed to test the null hypothesis that Cronbach s alpha reliability for the average of two raters is equal to.7 with power =.90, α =.05, and a reliability planning value of.8? 36. How large of a sample is needed to obtain a 95% confidence interval for Cronbach s alpha reliability of a scale with 6 items with a lower planning limit of.7 and an upper planning limit of.9? 37. Suppose a researcher obtained a 95% confidence interval for μ using a first-stage sample of n = 20 and obtained a confidence interval width of 6.4. How many participants should be sampled in the second stage to reduce the 95% confidence interval width to 4.0? 38. Suppose a researcher obtained a 95% confidence interval for a difference in two correlations in a 2-group design using a first-stage sample per group of 50 and obtained a confidence interval width of 0.4. How many participants should be sampled per group in the second stage to reduce the 95% confidence interval width to 0.3? 39. Suppose a researcher obtained a 95% confidence interval for φ in a within-subjects design using 40 participants in a first-stage sample and obtained a confidence interval width of 1.1. How many participants should be sampled in the second stage to reduce the 95% confidence interval width to 0.75? 40. For a 2-group design, what is the sample size requirement per group to obtain a 95% confidence interval for μ 1 μ 2 with a width of 8.0, a variance planning value of 50, and a sample size in group 2 that is twice as large as the sample size in group 1? 41. How large of a sample is needed to obtain a 95% confidence interval for slope coefficient in a fixed-x model with a desired confidence interval width of 2, σ x 2 = 10, and a within-group variance planning value of 80? 42. How large of a sample is needed to test H0: β 1 = 1 in a fixed-x model with a desired power of.9, σ x 2 = 25, and a within-group variance planning value of 250?

7 43. How large of a sample is needed to obtain a 95% confidence interval for residual variance that has an upper to lower limit ratio of 1.5 in a linear model with one predictor variable? 44. What sample size is required in a sign test of H0: τ = 50 with power =.9, α =.05, and π =.75? 45. What sample size is required in a Mann-Whitney test of H0: π =.5 with power =.80, α =.05, and π =.7?

8 Answers to Selected Computational Problems