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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

- 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 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

- 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

- 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

- 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

- Using normal approximation to binomial distribution

- 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,…)

- Random samples from independent groups with normal distributions and known1 and 2
- Any linear combination (e.g. the difference) of normal distributions (k, k) is also normal

CLT:

Populations the same

- Random samples from independent groups with normal distributions, but with equal and unknown1 and 2
- Using the pooled sample variance

H0: µ1 = µ2

- Random samples from independent groups with normal distributions, with unequal and unknown1 and 2
- Using the Satterthwaiteapproximation to the degrees of freedom (df)
- Use Excel Data Analysis tool!

- 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)

- 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

- 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

- 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

- 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

- 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

- 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

- 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 = MSA / MSW
- Reject H0 if F > FU(,c-1,n-c) [or p<]
- FU from Table A.7

- One-Way ANOVA Summary

- 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

- 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

- 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

- Total variation
- Among-group variation
- Among-block variation

(Grand mean)

(Group mean)

(Block mean)

- 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

- 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 = MSA / MSE
- Reject H0 if F > FU(,c-1,(r-1)(c-1))
- FU from Table A.7

- Two-Way ANOVA Summary

- 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

- 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)

- 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)

- Work through Appendix 10.1
- Work and hand in Problems
- 10.27
- 10.28 (except part c)

- Read Chapter 11
- Design of Experiments: Factorial Designs