Statistics 11 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 12 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 13 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 14 Calculation: CI = M ± t s M And to use the CI CI = M - t s M < μ < M + t s M

Statistics 15 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 16 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 17 DATA: 13, 10, 8, 13, 9, 14, 12, 10, 11, 10, 15, 13, 7, 6, 15, 10 SS ? Var? S M ? df? 90% CI ? 95% CI ?

Statistics 18 Between Groups ANOVA Next step: Comparing three or more samples Nothing really new, just extending what is already learned

Statistics 19

10 For t-statistic: Single alternative hypothesis (H1) Nondirectional (two-tail) Directional (one-tail) For F-statistic: Many alternative hypotheses (H1's) Always nondirectional

Statistics 111 Design: Between Groups ANOVA Partition the total variance of a sample into two separate sources (hence name of test)

Statistics 112

Statistics 113 Partition the total variance of a sample into two separate sources (hence name of test) Total variance –Variance associated with treatments and error

Statistics 114 Total variance –Variance associated with treatments and error –Variance associated with just error

Statistics 115 Calculations: Between Groups ANOVA

Statistics 116 Treatment 1Treatment 2Treatment n= Σx= Σx ²= Treatment mean

Statistics 117 Computational Formula for SS BG SS BG = (ΣX 1 ) ² n 1 + (ΣX 2 ) ² n 2 + (ΣX 3 ) ² n 3..+ (ΣX k ) ² n k [ (ΣX T ) ² n T ]

Statistics 118 Computational Formula for SS W SS W = ΣX²ΣX² [ (ΣX 1 ) ² n 1 + (ΣX2)²n2(ΣX2)²n2 + (ΣX 3 ) ² n 3..+ (ΣX K ) ² n K ]

Statistics 119 ANOVA Summary SourceSSdfMSF-Ratio Treaments SS BG Error SS W Total SS Total

Statistics 120 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 121 Body of table typically gives values for p <.05 and p <.01 Reject null hypothesis if: Obtained value exceeds tabled value

Statistics 122 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 123 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 124 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 125 Number of different tests Fisher’s LSD Test (Least Significant Difference)

Statistics 126 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 127 Other Post –Hocs comparisions Scheffe Newman-Keuls Duncan Bonferroni