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Pooled Variance t Test Tests means of 2 independent populations having equal variances Parametric test procedure Assumptions – Both populations are normally distributed – If not normal, can be approximated by normal distribution (n 1 30 & n 2 30 ) – Population variances are unknown but assumed equal

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Two Independent Populations Examples An economist wishes to determine whether there is a difference in mean family income for households in 2 socioeconomic groups. An admissions officer of a small liberal arts college wants to compare the mean SAT scores of applicants educated in rural high schools & in urban high schools.

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Pooled Variance t Test Example Youre a financial analyst for Charles Schwab. You want to see if there a difference in dividend yield between stocks listed on the NYSE & NASDAQ. NYSE NASDAQ Number 2125 Mean3.272.53 Std Dev1.301.16 Assuming equal variances, is there a difference in average yield ( =.05)? © 1984-1994 T/Maker Co.

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Pooled Variance t Test Solution H 0 : 1 - 2 = 0 ( 1 = 2 ) H 1 : 1 - 2 0 ( 1 2 ).05 df 21 + 25 - 2 = 44 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at =.05 There is evidence of a difference in means

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Test Statistic Solution

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Youre a research analyst for General Motors. Assuming equal variances, is there a difference in the average miles per gallon (mpg) of two car models ( =.05)? You collect the following: Sedan Van Number1511 Mean22.0020.27 Std Dev4.77 3.64 Pooled Variance t Test Thinking Challenge AloneGroupClass

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Solution Template H 0 : H 1 : = df = Critical Value(s): Test Statistic: Decision: Conclusion:

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Test Statistic Solution*

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One-Way ANOVA F-Test

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2 & c-Sample Tests with Numerical Data

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Experiment Investigator controls one or more independent variables – Called treatment variables or factors – Contain two or more levels (subcategories) Observes effect on dependent variable – Response to levels of independent variable Experimental design: Plan used to test hypotheses

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Completely Randomized Design Experimental units (subjects) are assigned randomly to treatments – Subjects are assumed homogeneous One factor or independent variable – 2 or more treatment levels or classifications Analyzed by: – One-Way ANOVA – Kruskal-Wallis rank test

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Randomized Design Example

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One-Way ANOVA F-Test Tests the equality of 2 or more (c) population means Variables – One nominal scaled independent variable 2 or more (c) treatment levels or classifications – One interval or ratio scaled dependent variable Used to analyze completely randomized experimental designs

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One-Way ANOVA F-Test Assumptions Randomness & independence of errors – Independent random samples are drawn Normality – Populations are normally distributed Homogeneity of variance – Populations have equal variances

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One-Way ANOVA F-Test Hypotheses H 0 : 1 = 2 = 3 =... = c – All population means are equal – No treatment effect H 1 : Not all j are equal – At least 1 population mean is different – Treatment effect – 1 2... c is wrong

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Compares 2 types of variation to test equality of means Ratio of variances is comparison basis If treatment variation is significantly greater than random variation then means are not equal Variation measures are obtained by partitioning total variation One-Way ANOVA Basic Idea

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ANOVA Partitions Total Variation Variation due to treatment Variation due to random sampling Total variation Sum of squares within Sum of squares error Within groups variation Sum of squares among Sum of squares between Sum of squares model Among groups variation

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Total Variation X

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Among-Groups Variation X X 3 X 3 X 2 X 2 X 1 X 1

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Within-Groups Variation X 2 X 2 X 1 X 1 X 3 X 3

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One-Way ANOVA Test Statistic Test statistic – F = MSA / MSW MSA is Mean Square Among MSW is Mean Square Within Degrees of freedom – df 1 = c -1 – df 2 = n - c c = # Columns (populations, groups, or levels) n = Total sample size

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One-Way ANOVA Summary Table

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One-Way ANOVA Critical Value If means are equal, F = MSA / MSW 1. Only reject large F! Always One-Tail! © 1984-1994 T/Maker Co.

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One-Way ANOVA F-Test Example As production manager, you want to see if 3 filling machines have different mean filling times. You assign 15 similarly trained & experienced workers, 5 per machine, to the machines. At the.05 level, is there a difference in mean filling times? Mach1Mach2Mach3 25.4023.4020.00 26.3121.8022.20 24.1023.5019.75 23.7422.7520.60 25.1021.6020.40

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One-Way ANOVA F-Test Solution H 0 : 1 = 2 = 3 H 1 : Not all equal =.05 df 1 = 2 df 2 = 12 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at =.05 There is evidence pop. means are different =.05 =.05

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Summary Table Solution

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Summary Table Excel Output

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One-Way ANOVA Thinking Challenge Youre a trainer for Microsoft Corp. Is there a difference in mean learning times of 12 people using 4 different training methods ( =.05)? M1M2M3M4 10111318 916823 59925 © 1984-1994 T/Maker Co. AloneGroupClass

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One-Way ANOVA Solution Template H 0 : H 1 : = df 1 = df 2 = Critical Value(s): Test Statistic: Decision: Conclusion:

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Summary Table (Partially Completed)

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One-Way ANOVA Solution* H 0 : 1 = 2 = 3 = 4 H 1 : Not all equal =.05 df 1 = 3 df 2 = 8 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at =.05 There is evidence pop. means are different =.05 =.05

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Summary Table Solution*

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