3.7 (a)

\(H_o: \mu_1 = \mu_2 = \mu_3 = \mu_4\)

\(H_a\): At least one of the means are different

mixing technique 1: 2971

mixing technique 2: 3156.25

mixing technique 3: 2933.75

mixing technique 4: 2666.25

F-Value: 12.73 p-value: 0.000489

Therefore we reject the null hypothesis because the p-value(0.000489) is < alpha (0.05).

mt1 <- c(3129, 3000, 2865, 2890)
mt2 <- c(3200, 3300, 2975, 3150)
mt3 <- c(2800, 2900, 2985, 3050)
mt4 <- c(2600,2700,2600,2765)

combinedGroups <- data.frame(cbind(mt1,mt2,mt3,mt4))
stackedGroups <- stack(combinedGroups)
dat <- aov(values~ind, data = stackedGroups)
summary(dat)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## ind          3 489740  163247   12.73 0.000489 ***
## Residuals   12 153908   12826                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

3.10 (a)

\(H_o: \mu_1 = \mu_2 = \mu_3 = \mu_4 =\mu5\)

\(H_a\): At least on of the means are different

F- value = 14.76

P- value = 0.000000913

Therefore we reject the null hypothesis because the p-value(0.000000913) is < alpha (0.05).

cotton15 <- c(7,7,15,11,9)
cotton20 <- c(12,17,12,18,18)
cotton25 <- c(14,19,19,18,18)
cotton30 <- c(19,25,22,19,23)
cotton35 <-c(7,10,11,15,11)

combinedGroups2 <- data.frame(cbind(cotton15,cotton20,cotton25,cotton30,cotton35))
stackedGroups2 <- stack(combinedGroups2)
dat2 <- aov(values~ind, data = stackedGroups2)
summary(dat2)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## ind          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

This means that at least one of the tensile strength for the various cotton differ from all the other means. The cotton effects the mean of the tensile strength ### 3.20 (a, b)

\(H_o: \mu_1 = \mu_2 = \mu_3 = \mu_4\)

\(H_a\): At least one of the means are different

lvl10 <- c(1530,1530,1440)
lvl15 <- c(1610, 1650, 1500)
lvl20 <- c(1560, 1730,1530)
lvl25 <- c(1500,1490,1510)

combinedGroups3 <- data.frame(cbind(lvl10, lvl15, lvl20, lvl25))
stackedGroups3 <- stack(combinedGroups3)
dat3 <- aov(values~ind, data = stackedGroups3)
summary(dat3)
##             Df Sum Sq Mean Sq F value Pr(>F)
## ind          3  28633    9544   1.865  0.214
## Residuals    8  40933    5117

The p-value(0.214) is greather than alpha(0.05). There for we fail to reject and accept the null hypothesis. There is not significant enough evidance to say that the rodding levels effects of the compressive strength.

The F-statisic is: 1.865