Lecture slides stats1.13.l10.air

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Lecture slides stats1.13.l10.air

  1. 1. Statistics One Lecture 10 Confidence intervals 1
  2. 2. Two segments •  Confidence intervals for sample means (M) •  Confidence intervals for regression coefficients (B) 2
  3. 3. Lecture 10 ~ Segment 1 Confidence intervals for sample means (M) 3
  4. 4. Confidence intervals •  All sample statistics, for example, a sample mean (M), are point estimates •  More specifically, a sample mean (M) represents a single point in a sampling distribution 4
  5. 5. The family of t distributions 5
  6. 6. Confidence intervals •  The logic of confidence intervals is to report a range of values, rather than a single value •  In other words, report an interval estimate rather than a point estimate 6
  7. 7. Confidence intervals •  Confidence interval: an interval estimate of a population parameter, based on a random sample –  Degree of confidence, for example 95%, represents the probability that the interval captures the true population parameter 7
  8. 8. Confidence intervals •  The main argument for interval estimates is the reality of sampling error •  Sampling error implies that point estimates will vary from one study to the next •  A researcher will therefore be more confident about accuracy with an interval estimate 8
  9. 9. Confidence intervals for M •  Example, IMPACT •  Assume N = 30 and multiple samples… –  Symptom Score (Baseline), M = 0.05 –  Symptom Score (Baseline), M = 0.07 –  Symptom Score (Baseline), M = 0.03 –  … 9
  10. 10. Confidence intervals for M •  Example, IMPACT •  Assume N = 10 and multiple samples… –  Symptom Score (Baseline), M = 0.01 –  Symptom Score (Baseline), M = 0.20 –  Symptom Score (Baseline), M = 0.00 –  … 10
  11. 11. Confidence intervals for M •  Example, IMPACT •  Assume N = 30 and multiple samples… –  Symptom Score (Post-injury), M = 12.03 –  Symptom Score (Post-injury), M = 12.90 –  Symptom Score (Post-injury), M = 14.13 –  … 11
  12. 12. Confidence intervals for M •  Example, IMPACT •  Assume N = 10 and multiple samples… –  Symptom Score (Post-injury), M = 19.70 –  Symptom Score (Post-injury), M = 8.40 –  Symptom Score (Post-injury), M = 13.30 –  … 12
  13. 13. Confidence intervals for M •  The width of a confidence interval is influenced by –  Sample size –  Variance in the population (and sample) –  Standard error (SE) = SD / SQRT(N) 13
  14. 14. Confidence intervals for M •  Example, IMPACT •  Assume N = 30 –  Symptom Score (Baseline) –  95% confidence interval •  -0.03 – 0.10 14
  15. 15. Confidence intervals for M •  Example, IMPACT •  Assume N = 10 –  Symptom Score (Baseline) –  95% confidence interval •  -0.10 – 0.50 15
  16. 16. Confidence intervals for M •  Example, IMPACT •  Assume N = 30 –  Symptom Score (Post-injury) –  95% confidence interval •  7.5 – 18.3 16
  17. 17. Confidence intervals for M •  Example, IMPACT •  Assume N = 10 –  Symptom Score (Post-injury) –  95% confidence interval •  2.7 – 23.9 17
  18. 18. Confidence interval for M •  Upper bound = M + t(SE) •  Lower bound = M – t(SE) •  SE = SD / SQRT(N) •  t depends on level of confidence desired and sample size 18
  19. 19. The family of t distributions 19
  20. 20. Segment summary •  All sample statistics, for example, a sample mean (M), are point estimates •  More specifically, a sample mean (M) represents a single point in a sampling distribution 20
  21. 21. Segment summary •  The logic of confidence intervals is to report a range of values, rather than a single value •  In other words, report an interval estimate rather than a point estimate 21
  22. 22. Segment summary •  The width of a confidence interval is influenced by –  Sample size –  Variance in the population (and sample) –  Standard error (SE) = SD / SQRT(N) 22
  23. 23. END SEGMENT 23
  24. 24. Lecture 10 ~ Segment 2 Confidence intervals for regression coefficients (B) 24
  25. 25. Confidence intervals for B •  All sample statistics, for example, a regression coefficient (B), are point estimates •  More specifically, a regression coefficient (B) represents a single point in a sampling distribution 25
  26. 26. Confidence intervals for B •  In regression, t = B / SE 26
  27. 27. The family of t distributions 27
  28. 28. Confidence intervals for B •  The logic of confidence intervals is to report a range of values, rather than a single value •  In other words, report an interval estimate rather than a point estimate 28
  29. 29. Confidence intervals for B •  The main argument for interval estimates is the reality of sampling error •  Sampling error implies that point estimates will vary from one study to the next •  A researcher will therefore be more confident about accuracy with an interval estimate 29
  30. 30. Confidence intervals for B •  The width of a confidence interval is influenced by –  Sample size –  Variance in the population (and sample) –  Standard error (SE) = SD / SQRT(N) 30
  31. 31. Confidence intervals for B •  Example, IMPACT •  Assume N = 40 –  Visual memory = B0 + (B)Verbal memory 31
  32. 32. Confidence intervals for B
  33. 33. Confidence intervals for B •  Example, IMPACT •  Assume N = 20 –  Visual memory = B0 + (B)Verbal memory 33
  34. 34. Confidence intervals for B 34
  35. 35. Segment summary •  All sample statistics are point estimates •  More specifically, a sample mean (M) or a regression coefficient (B) represents a single point in a sampling distribution 35
  36. 36. Segment summary •  The logic of confidence intervals is to report a range of values, rather than a single value •  In other words, report an interval estimate rather than a point estimate 36
  37. 37. Segment summary •  The width of a confidence interval is influenced by –  Sample size –  Variance in the population (and sample) –  Standard error (SE) = SD / SQRT(N) 37
  38. 38. END SEGMENT 38
  39. 39. END LECTURE 10 39

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