This report captures work done for the individual homework for Week 4. R code along with the results are provided. The required homework problems were taken from “Design and Analysis of Experiments 8th Edition”:
   1) 3.7a
   2) 3.10a
   3) 3.20a, 3.20b
Answers to the questions are in blue.

Problem 1 (3.7a)

QUESTION

The tensile strength of Portland cement is being studied. Four different mixing techniques can be used economically. A completely randomized experiment was conducted and the following data were collected: [See table of values in book.]
  (a) Test the hypothesis that mixing techniques affect the strength of the cement.
    Use alpha = 0.05

PROBLEM SETUP

Statistical Test and Rationale:  One-way ANOVA
Hypothesis Test:        Null:     H0:μ1 = μ2 = μ3 = μ4
               Alternate: H1:μi ≠ μj for at least one pair (i,j)
               Where μ1 -> μ4 = tensile strength in (lb/sq. inch) for 4 different mixing techniques.
One or Two Sided:      One-sided
Alpha:            0.05



STATISTICAL TEST

summary(aov(Tensiles ~ Groups, data= Techniques))
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## Groups       3 489740  163247   12.73 0.000489 ***
## Residuals   12 153908   12826                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

CONCLUSION

(a & b) The p-value above is < 0.0005, and with an alpha of 0.05, we reject the Null hypothesis. There is evidence to support the claim of a significant difference between the sample means of the various mixing techniques on tensile strength in Portland cement.

Problem 2 (3.10a)

QUESTION

A product developer is investigating the tensile strength of a new synthetic fiber that will be used to make cloth for men’s shirts. Strength is usually affected by the percentage of cotton used in the blend of materials for the fiber. The negineer conducts a completely randomized experiment with five levels of cotton content and replicates the experiment five times. The data are shown in the following table. [data table provided in book]
  (a) Is there evidence to support the claim that cotton content affects the mean tensile strength? Use alpha = 0.05.

PROBLEM SETUP

Statistical Test and Rationale:  One-way ANOVA
Hypothesis Test:        Null:     H0:μ1 = μ2 = μ3 = μ4 = μ5
               Alternate: H1:μi ≠ μj for at least one pair (i,j)
               Where μ1 -> μ5 = tensile strength for the 5 different cotton weight percents.
One or Two Sided:      One-sided
Alpha:            0.05

STATISTICAL TEST

summary(aov(Tensiles ~ CottonWeightPercents, data= Techniques))
##                      Df Sum Sq Mean Sq F value   Pr(>F)    
## CottonWeightPercents  4  475.8  118.94   14.76 9.13e-06 ***
## Residuals            20  161.2    8.06                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

CONCLUSION

(a & b) The p-value above is < 0.00001, and with an alpha of 0.05, we reject the Null hypothesis. There is evidence to support the claim of a significant difference between the sample means of the various cotton weight percents on tensile strength.

Problem 3 (3.20a & 3.20b)

QUESTION

An article in the ACI Materials Journal (Vol. 84, 1987, pp. 213-216) describes several experiments investigating the rodding of concrete to remove entrapped air. A 3-inch x 6-inch cylinder was used, and the number of times this rod was used is the design variable. The resulting compressive strength of the concrete specimen is the response. The data are shown in the following table: [data table provided in book]
  (a) Is there any difference in compressive strength due to the rodding level? Use alpha = 0.05.
  (b) Find the P-value for the F statistic in part (a).

PROBLEM SETUP

Statistical Test and Rationale:  One-way ANOVA
Hypothesis Test:        Null:     H0:μ1 = μ2 = μ3 = μ4
               Alternate: H1:μi ≠ μj for at least one pair (i,j)
               Where μ1 -> μ4 = compressive strength for the 4 different rodding levels.
One or Two Sided:      One-sided
Alpha:            0.05

STATISTICAL TEST

summary(aov(CompressiveStrengths ~ RoddingLevels, data= Techniques))
##               Df Sum Sq Mean Sq F value Pr(>F)
## RoddingLevels  3  28633    9544   1.865  0.214
## Residuals      8  40933    5117

CONCLUSION

(a & b) The p-value above is 0.214, and with an alpha of 0.05, we fail to reject the Null hypothesis and with the information given, we can only conclude that the rodding level doesn’t impact the compressive strength of the concrete.