Question

Use calculator or Excel to solve the following problems. DO NOT USE MINITAB (except e). Do NOT use Excel built-in functions or solvers (except t.inv() and f.inv() to obtain critical t and f values).

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:

Technique	Tensile Strength	Tensile Strength	Tensile Strength	Tensile Strength
1	3129	3000	2865	2890
2	3200	3300	2975	3150
3	2800	290	2985	3050
4	2600	2700	2600	2765

a) Test the hypothesis that mixing techniques affect the strength of the cement (α = 0.05).

b) Use the Fisher LSD method with α = 0.05 to make comparisons between pairs of means.

c) Repeat part (b) using Tukey’s test. Do you get the same conclusions as part (b)? If not, explain why.

d) Find a 95 percent confidence interval on the mean tensile strength of the Portland cement produced by each of the four mixing techniques. Also find a 95 percent confidence interval on the difference in means for techniques a and c.

e) Construct a normal probability plot of the residuals and a plot of residuals vs. predicted values (using Minitab). What conclusion would you draw from each plot?

f) Suppose Technique d is the current mixing technique. We would like to compare it with the average effect of the new techniques (Techniques a, b and c). Also, we are interested in whether Technique b is significantly different from the average effect of the two other new techniques (Techniques a and c). Construct the required contrasts to test these and apply Scheffe’s method to make conclusions. Set α = 0.05

Answer 1

Hypothesis:

H0: The mean strength of the cement is the same for 4 types of technique.
H1: The mean strength of the cement is not the same for 4 types of technique.

Let a represent the mixing strength of technique 1

=(3129   3000   2865   2890)
Let b represent the mixing strength of technique 2

= (3200   3300   2975   3150)
Let c represent the mixing strength of technique 3

= (2800   2900   2985   3050)

Let c represent the mixing strength of technique 4

= (2600   2700   2600   2765)
(g represents grand total)
We find the mean of both groups and the entire data

No of observation in each group

no. of groups = k = 4

Next we find the difference between each observation in the group with mean of the group and square it up.
example In group A we take the first observation and minus it the group mean ,this is done for each observation and for every group and summed up

Sum of square within the groups

Mean sum of square within the groups
MSW = rac{SSW}{ (n_g-k)}= 163246.72

Sum of square between the groups

Mean sum of square between the groups

Using the F table we find pvalue with df1 = 3 and df2=12
pvalue = 0.0004887

Table summarizing the output

Since the pvalue is less than 0.05, we reject the null hypothesis and conclude that the mean strength of the cement is not same for 4 types of technique

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