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:
# Load data
dat37<-read.csv("https://raw.githubusercontent.com/forestwhite/RStatistics/main/37Table.csv", header=TRUE)
dat37## X1 X2 X3 X4
## 1 3129 3200 2800 2600
## 2 3000 3300 2900 2700
## 3 2865 2975 2985 2600
## 4 2890 3150 3050 2765(a) Test the hypothesis that mixing techniques affect the strength of the cement. Use α = 0.05.
Lookup the P-value for f = 12.72811, degrees of freedom: Treatment = 3, degrees of freedom: Error = 12, and α = 0.05.
## calculate mean square error (mean of residuals), mean square of treatments
aov37 <- aov(values ~ ind,data=(stack(dat37)))
summary(aov37)## 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 ' ' 1P = 0 .000489 < 0.05 = p
∴ we can reject the null hypothesis and therefore there is sufficient evidence to support the hypothesis that mixing techniques affect the strength of cement using this data.
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 engineer 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.
# Load data
dat310<-read.csv("https://raw.githubusercontent.com/forestwhite/RStatistics/main/310Table.csv", header=TRUE)
dat310## X15 X20 X25 X30 X35
## 1 7 12 14 19 7
## 2 7 17 19 25 10
## 3 15 12 19 22 11
## 4 11 18 18 19 15
## 5 9 18 18 23 11(a) Is there evidence to support the claim that cotton content affects the mean tensile strength? Use α = 0.05.
Lookup the P-value for f = 14.75682, degrees of freedom: Treatment = 4, degrees of freedom: Error = 20, and α = 0.05.
## calculate mean square error (mean of residuals), mean square of treatments
aov310 <- aov(values ~ ind,data=(stack(dat310)))
summary(aov310)## 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 ' ' 1P = 0. 00000913 < 0.05 = p
∴ we reject the null hypothesis (means are equal) and there is evidence to support the claim that cotton content affects the mean tensile strength using this data.
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:
# Load data
dat320<-read.csv("https://raw.githubusercontent.com/forestwhite/RStatistics/main/320Table.csv", header=TRUE)
dat320## X10 X15 X20 X25
## 1 1530 1610 1560 1500
## 2 1530 1650 1730 1490
## 3 1440 1500 1530 1510(a) Is there any difference in compressive strength due to the rodding level? Use α = 0.05.
Lookup the P-value for f = 1.865364, degrees of freedom: Treatment = 3, degrees of freedom: Error = 8, and α = 0.05.
## calculate mean square error (mean of residuals), mean square of treatments
aov320 <- aov(values ~ ind,data=(stack(dat320)))
summary(aov320)## Df Sum Sq Mean Sq F value Pr(>F)
## ind 3 28633 9544 1.865 0.214
## Residuals 8 40933 5117P = 0.214 > 0.05 = p
∴ we cannot reject the null hypothesis and there is no distinguishable difference in compressive strength due to rodding level using this data.
(b) Find the P-value for the F statistic in part (a).
P = 0.214