During the cotton-spinning process, a strand of cotton thicker than the final thread is produced, and then twisted before it is wound onto a bobbin. The researcher wanted to investigate the process, and devised six different treatments (here labelled A, B, ..., F) which they wished to compare. An experimental unit consisted of the thread on the set of full bobbins in a machine on a given day. The response variable is the number of breaks per hundred pounds of material (named as \rate" in the dataset). An individual block is a machine with a single operator.

I have the following data:

rate,treat,block

4.2,A,1

4.6,A,2

5,A,3

4.1,A,4

5.5,A,5

3.2,A,6

10.1,A,7

4.2,A,8

5.1,A,9

4.6,A,10

11.5,A,11

5,A,12

6.4,A,13

6.4,B,1

8.3,B,2

7.9,B,3

8.8,B,4

10.1,B,5

11.5,B,6

8.7,B,7

9.7,B,8

8.3,B,9

9.2,B,10

9.2,B,11

10.1,B,12

12.4,B,13

2.3,C,1

3.3,C,2

7.3,C,3

10.6,C,4

7.9,C,5

5.5,C,6

7.8,C,7

5,C,8

7.8,C,9

6.4,C,10

8.3,C,11

9.2,C,12

12,C,13

3.3,D,1

6.4,D,2

4.1,D,3

6.9,D,4

6,D,5

7.4,D,6

6,D,7

7.3,D,8

7.8,D,9

7.4,D,10

7.3,D,11

10.1,D,12

7.8,D,13

3.7,E,1

6.4,E,2

8.3,E,3

3.3,E,4

7.8,E,5

5.9,E,6

8.3,E,7

5.1,E,8

6,E,9

3.7,E,10

11.5,E,11

13.8,E,12

8.3,E,13

6,F,1

9.7,F,2

7.4,F,3

11.5,F,4

17.9,F,5

11.9,F,6

10.2,F,7

7.8,F,8

10.6,F,9

17.5,F,10

10.6,F,11

10.6,F,12

8.7,F,13

I ran the analysis using the Randomised Complete Block Design (RCBD). I got the following output.

Randomised Complete Block Design (RCBD)

Df Sum Sq Mean Sq F value Pr(>F)

Treatment 2 813.8 406.9 59.688 <2e-16 ***

Block 77 984.2 12.8 1.875 0.000504 ***

Residuals 154 1049.9 6.8

Q1. The research is interested in conducting an additional experiment that will be able to find any of the existing treatments whose true difference in means is greater than 2 rates (the dependent variable) (e.g., the researcher would like the analysis to show a significant difference between üA and üB). Using “conventional" risks of making errors, how many blocks he would need? The researcher thinks that the variability in data will be similar to the results in the present experiment. Please look at the data. Since the mean of A when there are 11 blocks is 11.5, and most mean of B is less than 9.5, can i say he would need 11 blocks?

Q2. It is not feasible to use more than 24 blocks so, if 24 is not enough, what difference in means does the researcher have a good chance of detecting with 24 blocks? If the mean difference is greater tham 2 when there are 11 blocks, can I say that true mean difference is greater than 4.36 to detect with 24 blocks?

I have the following data:

rate,treat,block

4.2,A,1

4.6,A,2

5,A,3

4.1,A,4

5.5,A,5

3.2,A,6

10.1,A,7

4.2,A,8

5.1,A,9

4.6,A,10

11.5,A,11

5,A,12

6.4,A,13

6.4,B,1

8.3,B,2

7.9,B,3

8.8,B,4

10.1,B,5

11.5,B,6

8.7,B,7

9.7,B,8

8.3,B,9

9.2,B,10

9.2,B,11

10.1,B,12

12.4,B,13

2.3,C,1

3.3,C,2

7.3,C,3

10.6,C,4

7.9,C,5

5.5,C,6

7.8,C,7

5,C,8

7.8,C,9

6.4,C,10

8.3,C,11

9.2,C,12

12,C,13

3.3,D,1

6.4,D,2

4.1,D,3

6.9,D,4

6,D,5

7.4,D,6

6,D,7

7.3,D,8

7.8,D,9

7.4,D,10

7.3,D,11

10.1,D,12

7.8,D,13

3.7,E,1

6.4,E,2

8.3,E,3

3.3,E,4

7.8,E,5

5.9,E,6

8.3,E,7

5.1,E,8

6,E,9

3.7,E,10

11.5,E,11

13.8,E,12

8.3,E,13

6,F,1

9.7,F,2

7.4,F,3

11.5,F,4

17.9,F,5

11.9,F,6

10.2,F,7

7.8,F,8

10.6,F,9

17.5,F,10

10.6,F,11

10.6,F,12

8.7,F,13

I ran the analysis using the Randomised Complete Block Design (RCBD). I got the following output.

Randomised Complete Block Design (RCBD)

Df Sum Sq Mean Sq F value Pr(>F)

Treatment 2 813.8 406.9 59.688 <2e-16 ***

Block 77 984.2 12.8 1.875 0.000504 ***

Residuals 154 1049.9 6.8

Q1. The research is interested in conducting an additional experiment that will be able to find any of the existing treatments whose true difference in means is greater than 2 rates (the dependent variable) (e.g., the researcher would like the analysis to show a significant difference between üA and üB). Using “conventional" risks of making errors, how many blocks he would need? The researcher thinks that the variability in data will be similar to the results in the present experiment. Please look at the data. Since the mean of A when there are 11 blocks is 11.5, and most mean of B is less than 9.5, can i say he would need 11 blocks?

Q2. It is not feasible to use more than 24 blocks so, if 24 is not enough, what difference in means does the researcher have a good chance of detecting with 24 blocks? If the mean difference is greater tham 2 when there are 11 blocks, can I say that true mean difference is greater than 4.36 to detect with 24 blocks?

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