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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

ENGR 610Applied StatisticsFall 2007 - Week 8

Marshall University

CITE

Jack Smith

overview for today
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 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
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
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
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
Z Test on Proportion
  • Using normal approximation to binomial distribution
p value
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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 variation1
Partitioning of Total Variation
  • Total variation
  • Among-group variation
  • Among-block variation

(Grand mean)

(Group mean)

(Block mean)

partitioning cont d
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 variances1
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 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 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
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 means1
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
Homework
  • 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