Prof. Fazal Rehman Shamil

Question # 1:   Mark the following as True or False

1. Quantitative data can be converted into qualitative data.
2. Measures of central tendency include Median, Mode and Range.
3. An Indicator variable is a dummy variable used to distinguish between two categorical effects
4. SPSS stands for Statistical Package for Scientific studies.
5. A hierarchical or nested experimental design in one in which all possible combinations are included.
6. A Fractional factorial design is one in which no replications are made.
7. Questionnaires are usually analyzed through full factorial designs.
8. If there are four factors in a study and each factor has three levels, the minimum possible observations for a full factorial design will be 16.
9. Multiple regression is used when we want to regress more than one dependent variables on the same input variable.
10. If different variables have different levels, the design is termed as unbalanced.
11. Blocking is used to analyze the effect of a nuisance factor.
12. Replication is a term used to represent repeated duplication in an experiment.
13. If we need high resolution, we need to enhance the number of replications.
14. Critical F-statistic is the ratio of (1) sum of squares of treatment and (2) sum of squares of error.

Question # 2 Fill in the blanks in the following parts.

1. In a single factor problem with three levels and five replications per level, the degrees of freedom for the treatment are _______ while the degrees of freedom for error are ___________.
2. In a single factor problem, there are 3 replications of level 1, 5 replications of level 2 and 9 replications of level 3. The total number of observations will be _______, the treatment dof are __________ while the error dof are ___________.
3. We use pooled variance if the variances of individual groups are ____ .
4. When we want to prove a statement, we place that as _________ hypothesis (null / alternative/design/expected).

Question # 3:The following observations were recorded in an experiment involving six (A, B, C, D, E and F)

factors, each with two levels (-1 and +1).  A quarter fraction design was decided.   For this purpose, the Generators I=ABCE and I=BCDF were used.

1. Determine the third generator.
2. Determine which interactions are aliased (confounded) with Factor A and which ones are confounded with factor B.
3. Fill in the four blank columns in the table given below.
4. ANOVA Table is provided below the tabulated data. State which factors are significant and which ones are not.

Two generators (each with four factors)  were used, one involving E and the other F.  Determine the two generators and the resulting third generator.    Also figure out which factors or interactions are aliased.

 Run A B C D E F ABC ABD ACD BCD observation 1 -1 -1 -1 -1 -1 -1 6 2 1 -1 -1 -1 1 -1 10 3 -1 1 -1 -1 1 1 32 4 1 1 -1 -1 -1 1 60 5 -1 -1 1 -1 1 1 4 6 1 -1 1 -1 -1 1 15 7 -1 1 1 -1 -1 -1 26 8 1 1 1 -1 1 -1 60 9 -1 -1 -1 1 -1 1 8 10 1 -1 -1 1 1 1 12 11 -1 1 -1 1 1 -1 34 12 1 1 -1 1 -1 -1 60 13 -1 -1 1 1 1 -1 16 14 1 -1 1 1 -1 -1 5 15 -1 1 1 1 -1 1 37 16 1 1 1 1 1 1 52

A        1   770.06   770.06   770.06   22.75  0.002

B        1  5076.56  5076.56  5076.56  149.98  0.000

C        1     3.06     3.06     3.06    0.09  0.772

D        1     7.56     7.56     7.56    0.22  0.651

E        1     0.56     0.56     0.56    0.02  0.901

F        1     0.56     0.56     0.56    0.02  0.901

A*B      1   564.06   564.06   564.06   16.66  0.005

C*F      1     0.06     0.06     0.06    0.00  0.967

Error    7   236.94   236.94    33.85

Total   15  6659.44

Question # 4:   Suppose that a simpler model involving the factors A, B and their interaction (AB) was fitted on

the data of question 4 above.  The resulting model was

Compute the fitted values and the corresponding residuals (errors) for all the sixteen settings of the experiment.

Question # 5: Suppose that a company purchases raw material from three suppliers (1, 2, 3).  The company

wishes to determine if the purity of the raw material is the same from each supplier.  The company selects at random four batches from each supplier and runs two determinations of purity on each batch and then runs a Nested (or Hierarchical) design.

 Supplier 1 Supplier 2 Supplier 3 Batch 1 2 3 4 1 2 3 4 1 2 3 4 1 11 8 8 11 11 10 9 10 12 8 11 13 2 9 7 10 14 8 14 10 13 14 10 9 12

Table below provides expected mean squares in the two-stage nested design.

 A Fixed, B Fixed A Fixed, B random A Random, B Random
1. Point out which models should be used for determination of , and Point out which factor is nested within the other. In another word which would, you treat as A and which factor as B.
2. What will be the degrees of freedom for (1) Suppliers, (2) Batches, (3) Interactions, and (4) Error?

Question # 6:  For the tabulated data of question 6 (above), fill the following columns for entry in some software package.

 S No Supplier Batch Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Question # 7:   Attempt any three of the following:

1. Differentiate between fixed effect and random effect models.
2. What is randomization and why is it important?
3. Why is it important to ensure that the residuals follow Normal Distribution?
4. What are the assumptions in Analysis of variance and regression analysis.
5. What are the sampling distributions? Prof.Fazal Rehman Shamil (Available for Professional Discussions)
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