Question

3. The tensile strength of Portland cement is being studied. Four different mixing techniques can be used economically. The following data have been collected: Technique Tensile strength 3129 3000 2865 2890 3200 3300 2975 3150 2800 2900 2985 3050 2600 2700 2600 2765 3 and can be read into R with the commands: y c(3129, 3000, 2865, 2890, 3200, 3300 , 2975, 3150, 2800, 2900, 2985, 3050 2600 , 2700 , 2600 2765 x = c(rep(1,4),rep (2,4),rep (3,4),rep(4,4)) Test the hypothesis that mixing techniques affect the strength of the cement. Use the one-way ANOVA model, assuming that the assumptions are met. Report the ANOVA table, including the p-value, and your conclusion with α = .05.

Answer 1

We use one way analysis of variance to test the hypothesis that mixing techniques affect the tensile strength of cement.

ONE WAY ANOVA TABLE:

HYPOTHESIS:

The hypothesis is given by:

$H_{0}:$ The mean tensile strength of cement for four different mixing techniques are the same.

$H_{1}:$ The mean tensile strength of cement for at least two different mixing techniques differ.

Since the p-value is 0.000489 which is less than 0.05. Hence we reject $H_{0}$ at 5% level and conclude that the mean tensile strength for at least two different mixing techniques differ.

PAIRWISE COMPARISON:

We have found out that the mean tensile strength for at least two different mixing techniques differ. Thus we use tukey pairwise comparison to find between which two mixing techniques does the tensile strength of cement differs.

Since the p-value is less than 0.05, there is a statistically significant difference in the mean tensile strength of cement between the mixing techniques 1 and 4, techniques 2 and 4, and techniques 3 and 4 at 5% level of significance.

• The difference in mean tensile strength of cement between the mixing techniques 1 and 4 is -304.75 (i.e.) the tensile strength of technique 1 is 304.75 units higher than technique 4. The 95% confidence interval is (-542.50,-66.997). That is, the difference in the mean tensile strength between the mixing techniques 1 and 4 may lie between -542.50 and -66.997.
• The difference in mean tensile strength of cement between the mixing techniques 2 and 4 is -490 (i.e.) the tensile strength of technique 2 is 490 units higher than technique 4. The 95% confidence interval is (-727.75,-252.25). That is, the difference in the mean tensile strength between the mixing techniques 2 and 4 may lie between --727.75 and -252.25.
• The difference in mean tensile strength of cement between the mixing techniques 3 and 4 is -267.50 (i.e.) the tensile strength of technique 3 is 267.50 units higher than technique 4. The 95% confidence interval is (-505.25,-29.75). That is, the difference in the mean tensile strength between the mixing techniques 3 and 4 may lie between -505.25 and -29.75.

BOX PLOT:

Comparison of Tensile Strength technique-1 technique-2 technique-3 technique-4 Technique

From the box plot we can conclude that there is a significant difference in the average tensile strength for different mixing techniques. It can also be observed that the mean tensile strength for mixing technique 2 is the highest and mixing technique 4 is the lowest. The mean tensile strength for mixing techniques 1 and 3 are almost similar.