# ENGR 610 Applied Statistics Fall 2007 - Week 8 - PowerPoint PPT Presentation

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ENGR 610 Applied Statistics Fall 2007 - Week 8. Marshall University CITE Jack Smith. Overview for Today. Review Hypothesis Testing , Ch 9 Go over homework problem: 9.69, 9.71, 9.74 Design of Experiment , Ch 10 One-Factor Experiments Randomized Block Experiments Homework assignment.

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ENGR 610 Applied Statistics Fall 2007 - Week 8

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## ENGR 610Applied StatisticsFall 2007 - Week 8

Marshall University

CITE

Jack Smith

### Overview for Today

• Review Hypothesis Testing, Ch 9

• Go over homework problem: 9.69, 9.71, 9.74

• Design of Experiment, Ch 10

• One-Factor Experiments

• Randomized Block Experiments

• Homework assignment

### Critical Regions

• Critical value of test statistic (Z, t, F, 2,…)

• Based on desired level of significance ()

• Acceptance (of null hypothesis) region

• Rejection (alternative hypothesis) region

• Two-tailed or one-tailed

### Z Test ( known) - Two-tailed

• Critical value (Zc) based on chosen level of significance, 

• Typically  = 0.05 (95% confidence), where Zc = 1.96 (area = 0.95/2 = 0.475)

•  = 0.01 (99%) and 0.001 (99.9%) are also common, where Zc = 2.57 and 3.29

• Null hypothesis rejected if sample Z > Zc or < -Zc, where

### Z Test ( known) - One-tailed

• Critical value (Zc) based on chosen level of significance, 

• Typically  = 0.05 (95% confidence), but where Zc = 1.645 (area = 0.95 - 0.50 = 0.45)

• Null hypothesis rejected if sample Z > Zc, where

### t Test ( unknown) - Two-tailed

• Critical value (tc) based on chosen level of significance, , and degrees of freedom, n-1

• Typically  = 0.05 (95% confidence), where, for exampletc = 2.045 (upper area = 0.05/2 = 0.025), for n-1 = 29

• Null hypothesis rejected if sample t > tc or < -tc, where

t

### Z Test on Proportion

• Using normal approximation to binomial distribution

### p-value

• Use probabilities corresponding to values of test statistic (Z, t,…)

• Compare probability (p) directly to  instead of, say, t to tc

• If the p-value  , accept null hypothesis

• If the p-value < , reject null hypothesis

• Does not assume any particular distribution (Z-normal, t, F, 2,…)

### Z Test for the Difference between Two Means

• Random samples from independent groups with normal distributions and known1 and 2

• Any linear combination (e.g. the difference) of normal distributions (k, k) is also normal

CLT:

Populations the same

### t Test for the Difference between Two Means (Equal Variances)

• Random samples from independent groups with normal distributions, but with equal and unknown1 and 2

• Using the pooled sample variance

H0: µ1 = µ2

### t Test for the Difference between Two Means (Unequal Variances)

• Random samples from independent groups with normal distributions, with unequal and unknown1 and 2

• Using the Satterthwaiteapproximation to the degrees of freedom (df)

• Use Excel Data Analysis tool!

### F test for the Difference between Two Variances

• Based on F Distribution - a ratio of 2 distributions, assuming normal distributions

• FL(,n1-1,n2-1)  F  FU(,n1-1,n2-1), where FL(,n1-1,n2-1) = 1/FU(,n2-1,n1-1), and where FU is given in Table A.7 (using nearest df)

### Mean Test for Paired Data or Repeated Measures

• Based on a one-sample test of the corresponding differences (Di)

• Z Test for known population D

• t Test for unknown D (with df = n-1)

H0: D = 0

### 2 Test for the Difference among Two or More Proportions

• Uses contingency table to compute

• (fe)i = nip or ni(1-p) are the expected frequencies, where p = X/n, and (fo)i are the observed frequencies

• For more than 1 factor, (fe)ij = nipj, where pj = Xj/n

• Uses the upper-tail critical 2 value, with the df = number of groups – 1

• For more than 1 factor, df = (factors -1)*(groups-1)

Sum over all cells

### Other Tests

• 2 Test for the Difference between Variances

• Follows directly from the 2 confidence interval for the variance (standard deviation) in Ch 8.

• Very sensitive to non-Normal distributions, so not a robust test.

• Wilcoxon Rank Sum Test between Two Medians

### Design of Experiments

• R.A. Fisher (Rothamsted Ag Exp Station)

• Study effects of multiple factors simultaneously

• Randomization

• Homogeneous blocking

• One-Way ANOVA (Analysis of Variance)

• One factor with different levels of “treatment”

• Partitioning of variation - within and among treatment groups

• Generalization of two-sample t Test

• Two-Way ANOVA

• One factor against randomized blocks (paired treatments)

• Generalization of two-sample paired t Test

### One-Way ANOVA

• ANOVA = Analysis of Variance

• However, goal is to discern differences in means

• One-Way ANOVA = One factor, multiple treatments (levels)

• Randomly assign treatment groups

• Partition total variation (sum of squares)

• SST = SSA + SSW

• SSA = variation among treatment groups

• SSW = variation within treatment groups (across all groups)

• Compare mean squares (variances): MS = SS / df

• Perform F Test on MSA / MSW

• H0: all treatment group means are equal

• H1: at least one group mean is different

### Partitioning of Total Variation

• Total variation

• Within-group variation

• Among-group variation

(Grand mean)

(Group mean)

c = number of treatment groups

n = total number of observations

nj = observations for group j

Xij = i-th observation for group j

### Mean Squares (Variances)

• Total mean square (variance)

• MST = SST / (n-1)

• Within-group mean square

• MSW = SSW / (n-c)

• Among-group mean square

• MSA = SSA / (c-1)

### F Test

• F = MSA / MSW

• Reject H0 if F > FU(,c-1,n-c) [or p<]

• FU from Table A.7

• One-Way ANOVA Summary

### Tukey-Kramer Comparison of Means

• Critical Studentized range (Q) test

• qU(,c,n-c) from Table A.9

• Perform on each of the c(c-1)/2 pairs of group means

• Analogous to t test using pooled variance for comparing two sample means with equal variances

### One-Way ANOVA Assumptions and Limitations

• Assumptions for F test

• Random and independent (unbiased) assignments

• Normal distribution of experimental error

• Homogeneity of variance within and across group (essential for pooling assumed in MSW)

• Limitations of One-Factor Design

• Inefficient use of experiments

• Can not isolate interactions among factors

### Randomized Block Model

• Matched or repeated measurements assigned to a block, with random assignment to treatment groups

• Minimize within-block variation to maximize treatment effect

• Further partition within-group variation

• SSW = SSBL + SSE

• SSBL = Among-block variation

• SSE = Random variation (experimental error)

• Total variation: SST = SSA + SSBL + SSE

• Separate F tests for treatment and block effects

• Two-way ANOVA, treatment groups vs blocks, but the focus is only on treatment effects

### Partitioning of Total Variation

• Total variation

• Among-group variation

• Among-block variation

(Grand mean)

(Group mean)

(Block mean)

### Partitioning, cont’d

• Random error

c = number of treatment groups

r = number of blocks

n = total number of observations (rc)

Xij = i-th block observation for group j

### Mean Squares (Variances)

• Total mean square (variance)

• MST = SST / (rc-1)

• Among-group mean square

• MSA = SSA / (c-1)

• Among-block mean square

• MSBL = SSBL / (r-1)

• Mean square error

• MSE = SSE / (r-1)(c-1)

### F Test for Treatment Effects

• F = MSA / MSE

• Reject H0 if F > FU(,c-1,(r-1)(c-1))

• FU from Table A.7

• Two-Way ANOVA Summary

### F Test for Block Effects

• F = MSBL / MSE

• Reject H0 if F > FU(,r-1,(r-1)(c-1))

• FU from Table A.7

• Assumes no interaction between treatments and blocks

• Used only to examine effectiveness of blocking in reducing experimental error

• Compute relative efficiency (RE) to estimate leveraging effect of blocking on precision

### Estimated Relative Efficiency

• Relative Efficiency

• Estimates the number of observations in each treatment group needed to obtain the same precision for comparison of treatment group means as with randomized block design.

• nj (without blocking)  RE*r (with blocking)

### Tukey-Kramer Comparison of Means

• Critical Studentized range (Q) test

• qU(,c,(r-1)(c-1)) from Table A.9

• Where group sizes (number of blocks, r) are equal

• Perform on each of the c(c-1)/2 pairs of group means

• Analogous to paired t test for the comparison of two-sample means (or one-sample test on differences)

### Homework

• Work through Appendix 10.1

• Work and hand in Problems

• 10.27

• 10.28 (except part c)