Design and Analysis of Experiments

Take Home Work Exam 1Student: Nguyen Thi Hang Mark:

ID: R9605005Comment:

Problem.1. Conductivity measurements (µmho/cm) were taken at four different locations in the aerated lagoon of a pulp and paper mill. The lagoon is supposed to be mixed by aerators so the contents are homogeneous. The conductivity results were shown in the following table:

(1) Is the lagoon homogeneously mixed? Conduct an ANOVA for the data and draw your conclusions.

(2) Please check if the assumption of equal variances is satisfied.(3) Test all pairs of means using both the Turkey’s test and Fisher LSD test.

Please also compare the results from these two multiple comparison methods. What are your conclusions?

(4) Use the Kruskal-Wallis test for the experiment. And also compare the conclusions obtained with those from the usual ANOVA.

Table1 Conductivity measurements (µmho/cm) at the different locations on the aerated lagoon

-------------------------------------------------------------------------------------- Location A Location B Location C Location D------------------------------------------------------------------------------------620 630 680 560600 670 660 620630 710 710 600590 640 670 610

650 680 630660 680 640

630590

Solution.Conducting an ANOVA for the data we get below procedureWith hypothesis: H0: µ1 = µ2 = µ3 = µ4

H1: µi ≠ µj

Using Minitab software (chose with the following steps:

1. Input and prepare the data for the test

2. Select One-Way of ANOVA method

3. Set up the suitable parameters

we can use some options such as Graphs, Compatisons

Then, we can see the results from using the Minitab

Results and Discussion(1) Is the lagoon homogeneously mixed? Conduct an ANOVA for the data

and draw your conclusions.F0 = 13.57 > F0.05, 3, 20 = 3.10 or P = 0.000 < 0.05 thus we can conclude that H0 can be rejected or the lagoon is not homogeneously mixed.

(2) Please check if the assumption of equal variances is satisfied.

H0:

H1: ≠j

Use test for equal variances we get these results

To fix suitable parameters:

Both of Pvalue obtained (Pvalue = 0.644 and Pvalue = 0.593) are larger than 0.05 thus H0 can not rejected. It means all variances are the same.

(3) Test all pairs of means using both the Turkey’s test and Fisher LSD test. Please also compare the results from these two multiple comparison methods. What are your conclusions?

From

Above results indicate pairs of means that are significantly different for:A vs B, A vs C, B vs D, C vs D

And there are not significantly different for: A vs D, B vs C.

Above results indicate pairs of means that are significantly different for:A vs B, A vs C, B vs D, C vs D

And there are not significantly different for: A vs D, B vs C.

To compare the results from these two multipule comparison methods:These two method give the same results when we compare all pairs of means using both the Tukey’s test and Fisher ‘s test.

(4) Use the Kruskal-Wallis test for the experiment. And also compare the conclusions obtained with those from the usual ANOVA.

We have hypotheses: H0: µ1 = µ2 = µ3 = µ4

H1: µi ≠ µj

Above results indicate both of H = 16. 86 and H = 17.02 larger than or P = 0.001 < 0.05 we conclude to reject H0 or the lagoon is homogeneously mixed

Problem 2. The data below are from an experiment that attempted to examine the factors affecting the reaeration rate (y) in a laboratory model stream channel. The three experimental factors are stream velocity (V, in m/sec), stream depth (D, in cm), and channel roughness (R).

Table 2 Results of the reaeration rateRun V D R y

1 0.25 10 smooth 107 117 1172 0.5 10 Smooth 190 178 1793 0.25 15 smooth 119 116 1334 0.5 15 smooth 188 191 1955 0.25 10 smooth 119 132 1266 0.5 10 smooth 187 173 1667 0.25 15 smooth 140 133 1328 0.5 15 smooth 164 145 144

(1) Analyze the data and draw conclusions.(2) Analyze the residuals.

Solution

There are three factors V, D and R, each at two levels, are of interest. The design is called a 23 factorial design.

Using DOE method for 23 design to analyze the data:1. Input and prepare the data on Minitab

2. Set up suitable parameters.

3. Output the data

3. Results and Discussion(1) Analyze the data and draw conclusions.The ANOVA in the above Table is used to confirm the magnitude of these effects. From these Table we note that the main effect of V is highly significant (because has very small P-value). The V*D, V*R, D*R, and V*D*R are also highly significant; thus there are strong interaction between V and D, V and R; V, D, and R on the reaeration rate (y).

(2) Analyze the residuals.

From the normal probability plot of these residuals show that doesn’t reveal anything particularly troublesome.

Although the largest positive residual (13 at V = 0.05, D = 15, R = coarse) does stand out some what from others.The standardized value of this residual is = 1.69, and this is only residual whose absolute value is smaller than 2.

From plots the residuals versus the fitted values. This plots indicates that a mild tendency for the variance of the residuals to increase.