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Confidence Interval Suppose you have a sample from a population

Confidence Interval Suppose you have a sample from a population You know the sample mean is an unbiased estimate of population mean Question: What is the population mean? Answer: You will never really know .

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Confidence Interval Suppose you have a sample from a population

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  1. Confidence Interval • Suppose you have a sample from a population • You know the sample mean is an unbiased estimate of population mean • Question: What is the population mean? • Answer: You will never really know Statistics 1

  2. But ... you can determine, with some degree of certainty, a range which contains the mean • Range is called the Confidence Interval of the Mean Statistics 1

  3. Definition • A Confidence Interval is a statement concerning a range of values which is likely to include the population mean based upon a sample from the population. Statistics 1

  4. Calculation: • CI = M ± t sM • And to use the CI • CI = M - t sM < μ < M + t sM Statistics 1

  5. Some Important Notes: • For an interval estimate, you use a range of values as your estimate of an unknown quantity. • When an interval estimate is accompanied by a specific level of confidence (or probability), it is called a confidence interval. • The general goal of estimation is to determine how much effect a treatment has. Statistics 1

  6. The general goal of estimation is to determine how much effect a treatment has. • Whereas, the purpose of a confidence interval is to use a sample mean or mean difference to estimate the corresponding population mean or mean difference. • Also, for independent-measures t-statistics, the values used for estimation is the difference between two population means. Statistics 1

  7. DATA: • 13, 10, 8, 13, 9, 14, 12, 10, 11, 10, 15, 13, 7, 6, 15, 10 • SS ? • Var? • SM? • df? • 90% CI ? • 95% CI ? Statistics 1

  8. Between Groups ANOVA • Next step: Comparing three or more samples • Nothing really new, just extending what is already learned Statistics 1

  9. Statistics 1

  10. Statistics 1

  11. Design: Between Groups ANOVA • Partition the total variance of a sample into two separate sources (hence name of test) Statistics 1

  12. Statistics 1

  13. Partition the total variance of a sample into two separate sources (hence name of test) • Total variance • Variance associated with treatments and error Statistics 1

  14. Total variance • Variance associated with treatments and error • Variance associated with just error Statistics 1

  15. Calculations: Between Groups ANOVA Statistics 1

  16. Statistics 1

  17. Computational Formula for SSBG • SSBG = Statistics 1

  18. Computational Formula for SSW • SSW= Statistics 1

  19. Statistics 1

  20. Evaluating F-obtained: Between Groups ANOVA • Evaluate F-obtained value using an F-table • Similar to t-table except……… • Determining F value requires two separate degrees of freedom entries • Degrees of freedom for MS Between to locate the correct column • Degrees of freedom for MS Within to locate the correct row Statistics 1

  21. Body of table typically gives • values for p < .05 and p < .01 • Reject null hypothesis if: • Obtained value exceeds tabled value Statistics 1

  22. Formal Properties: Between Groups ANOVA • Between groups F-statistic is appropriate when • Independent measure is • Between subjects • Quantitative • Qualitative • Design includes three or more treatment groups • Dependent measure is • Quantitative • Scale of measurement is interval or better Statistics 1

  23. Between groups F-statistic assumes • Treatment groups are • Normally distributed • Homogeneity of within group variance • Subjects are: • Randomly and Independently selected from population • Randomly assigned to treatment groups Statistics 1

  24. Comparing Treatments: Between Groups ANOVA • Problem with multiple t-tests to compare treatment effects • Multiple t-tests would yield some significant decisions by chance • Can correct by making comparisons with a statistic that accounts for, "corrects for" multiple comparisons Statistics 1

  25. Number of different tests • Fisher’s LSD Test (Least Significant Difference) Statistics 1

  26. Tukey's HSD (Honest Significant Difference) • Where: • CD = Absolute critical difference • q = Studentized range value obtain from table entered with • k groups signifying appropriate column • df for within treatments MS signifying row • n = number of observations per group Statistics 1

  27. Other Post –Hocs comparisions • Scheffe • Newman-Keuls • Duncan • Bonferroni Statistics 1

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