Question 3.7

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 data were collected.

Part a. Test the hypothesis that mixing techniques affect the strength of the cement. Use a=0.05.

Null Hypothesis: Ho: mu1 = mu2 = mu3 = mu4 = mu

Alternative Hypothesis: Ha: Atleast one mui differs i = {1,2,3,4}

Reading Data

Pop1 <- c(3129, 3000, 2865, 2890)
Pop2 <- c(3200, 3300, 2975, 3150)
Pop3 <- c(2800, 2900, 2985, 3050)
Pop4 <- c(2600, 2700, 2600, 2765)
Pop <- rbind(Pop1, Pop2, Pop3, Pop4)

GA <- c(mean(Pop))

a <- mean(Pop1)
b <- mean(Pop2)
c <- mean(Pop3)
d <- mean(Pop4)

SSE, SSTr, MSE, MSTr, F Statistic & Critical Region Calculations

SSE1 <- (3129-a)^2 + (3000-a)^2 + (2865-a)^2 + (2890-a)^2
SSE2 <- (3200-b)^2 + (3300-b)^2 + (2975-b)^2 + (3150-b)^2
SSE3 <- (2800-c)^2 + (2900-c)^2 + (2985-c)^2 + (3050-c)^2
SSE4 <- (2600-d)^2 + (2700-d)^2 + (2600-d)^2 + (2765-d)^2

SSE <- SSE1 + SSE2 + SSE3 + SSE4

str(SSE)

##  num 153908

MSE <- SSE / (12)

str(MSE)

##  num 12826

SSTr <- 4*((a - GA)^2 + (b - GA)^2 + (c - GA)^2 + (d - GA)^2)

str(SSTr)

##  num 489740

MSTr <- SSTr / (3)

str(MSTr)

##  num 163247

SST <- SSE + SSTr

str(SST)

##  num 643648

Statistic <- MSTr / MSE

str(Statistic)

##  num 12.7

qf(0.95, 3, 12)

## [1] 3.490295

Answer Part a: Since our F-Statistic Value (12.728) is greater than critical value (3.4902), so we reject Ho and thus we conclude that mixing techniques do affect the strength of the cement.

Question 3.10

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.

Part a. Is there evidence to support the claim that cotton content affects the mean tensile strength? Use a=0.05.

Null Hypothesis: Ho: mu1 = mu2 = mu3 = mu4 = mu5 = mu

Alternative Hypothesis: Ha: Atleast one mui differs i = {1,2,3,4,5}

Reading Data

PopA <- c(7, 7, 15, 11, 9)
PopB <- c(12, 17, 12, 18, 18)
PopC <- c(14, 19, 19, 18, 18)
PopD <- c(19, 25, 22, 19, 23)
PopE <- c(7, 10, 11, 15, 11)

PopT <- rbind(PopA, PopB, PopC, PopD, PopE)

GA2 <- c(mean(PopT))

e <- mean(PopA)
f <- mean(PopB)
g <- mean(PopC)
h <- mean(PopD)
k <- mean(PopE)

SSE, SSTr, MSE, MSTr, F Statistic & Critical Region Calculations

SSEA <- (7-e)^2 + (7-e)^2 + (15-e)^2 + (11-e)^2 + (9-e)^2
SSEB <- (12-f)^2 + (17-f)^2 + (12-f)^2 + (18-f)^2 + (18-f)^2
SSEC <- (14-g)^2 + (19-g)^2 + (19-g)^2 + (18-g)^2 + (18-g)^2
SSED <- (19-h)^2 + (25-h)^2 + (22-h)^2 + (19-h)^2 + (23-h)^2
SSEE <- (7-k)^2 + (10-k)^2 + (11-k)^2 + (15-k)^2 + (11-k)^2

SSE <- SSEA + SSEB + SSEC + SSED + SSEE

str(SSE)

##  num 161

MSE <- SSE / (20)

str(MSE)

##  num 8.06

SSTr <- 5*((e - GA2)^2 + (f - GA2)^2 + (g - GA2)^2 + (h - GA2)^2 + (k - GA2)^2)

str(SSTr)

##  num 476

MSTr <- SSTr / (4)

str(MSTr)

##  num 119

SST <- SSE + SSTr

str(SST)

##  num 637

Statistic <- MSTr / MSE

str(Statistic)

##  num 14.8

qf(0.95, 4, 20)

## [1] 2.866081

Answer Part a: Since our F-Statistic Value (14.7568) is greater than critical value (2.86608), so we reject Ho and thus we conclude that cotton content does affect the mean tensile strength.

Question 3.20

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

Part a. Is there any difference in compressive strength due to the rodding level? Use a=0.05.

Null Hypothesis: Ho: mu1 = mu2 = mu3 = mu4 = mu

Alternative Hypothesis: Ha: Atleast one mui differs i = {1,2,3,4}

Reading Data

Popa <- c(1530, 1530, 1440)
Popb <- c(1610, 1650, 1500)
Popc <- c(1560, 1730, 1530)
Popd <- c(1500, 1490, 1510)

Popt <- rbind(Popa, Popb, Popc, Popd)

GA3 <- c(mean(Popt))

l <- mean(Popa)
m <- mean(Popb)
n <- mean(Popc)
p <- mean(Popd)

SSE, SSTr, MSE, MSTr, F Statistic & Critical Region Calculations

SSEa <- (1530-l)^2 + (1530-l)^2 + (1440-l)^2
SSEb <- (1610-m)^2 + (1650-m)^2 + (1500-m)^2
SSEc <- (1560-n)^2 + (1730-n)^2 + (1530-n)^2
SSEd <- (1500-p)^2 + (1490-p)^2 + (1510-p)^2

SSE <- SSEa + SSEb + SSEc + SSEd

str(SSE)

##  num 40933

MSE <- SSE / (8)

str(MSE)

##  num 5117

SSTr <- 3*((l - GA3)^2 + (m - GA3)^2 + (n - GA3)^2 + (p - GA3)^2)

str(SSTr)

##  num 28633

MSTr <- SSTr / (3)

str(MSTr)

##  num 9544

SST <- SSE + SSTr

str(SST)

##  num 69567

Statistic <- MSTr / MSE

str(Statistic)

##  num 1.87

qf(0.95, 3, 8)

## [1] 4.066181

Answer Part a: Since our F-Statistic Value (1.865) is less than critical value (4.066), so we fail to reject Ho and thus we conclude that there is no difference in compressive strength due to the rodding level.

Homework - Week 4

Muneeb

9/22/2021

Question 3.7

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 data were collected.

Part a. Test the hypothesis that mixing techniques affect the strength of the cement. Use a=0.05.

Null Hypothesis: Ho: mu1 = mu2 = mu3 = mu4 = mu

Alternative Hypothesis: Ha: Atleast one mui differs i = {1,2,3,4}

Reading Data

SSE, SSTr, MSE, MSTr, F Statistic & Critical Region Calculations

Answer Part a: Since our F-Statistic Value (12.728) is greater than critical value (3.4902), so we reject Ho and thus we conclude that mixing techniques do affect the strength of the cement.

Question 3.10

Part a. Is there evidence to support the claim that cotton content affects the mean tensile strength? Use a=0.05.

Null Hypothesis: Ho: mu1 = mu2 = mu3 = mu4 = mu5 = mu

Alternative Hypothesis: Ha: Atleast one mui differs i = {1,2,3,4,5}

Reading Data

SSE, SSTr, MSE, MSTr, F Statistic & Critical Region Calculations

Answer Part a: Since our F-Statistic Value (14.7568) is greater than critical value (2.86608), so we reject Ho and thus we conclude that cotton content does affect the mean tensile strength.

Question 3.20

Part a. Is there any difference in compressive strength due to the rodding level? Use a=0.05.

Null Hypothesis: Ho: mu1 = mu2 = mu3 = mu4 = mu

Alternative Hypothesis: Ha: Atleast one mui differs i = {1,2,3,4}

Reading Data

SSE, SSTr, MSE, MSTr, F Statistic & Critical Region Calculations

Answer Part a: Since our F-Statistic Value (1.865) is less than critical value (4.066), so we fail to reject Ho and thus we conclude that there is no difference in compressive strength due to the rodding level.

Part b. Find the P-value for the F statistic in part (a).

Answer Part b: By plugging in these statistics in a F Distribution Calculator we found the p-value = 0.2138.