**PlSc 724 - Practice Exam 2 Questions**

1. Assume you have ten wheat varieties and you plan a yield trial. List some of the points youshould consider in choosing a design. Possible designs are CRD, RCBD, and latin square. GOTO ANSWER 1.

2. Given that t=8 and r>1, which design (CRD, RCBD, LS) has the largest and which design theleast number of degrees of freedom for experimental error? GO TO ANSWER 2.

3. Tell how the randomization procedure for the CRD is different from the RCBD? GO TOANSWER 3.

4. Fill in the following diagram with possible randomizations for a CRD and RCBD Be sureyou assign treatments to the experimental units in such a way that the CRD cannot be mistakenfor a RCBD. T=5, r=3. GO TO ANSWER 4.

CRD

RCBD

5. In a RCBD, what is the purpose of grouping treatments into blocks? GO TO ANSWER 5. 6. In a RCBD, is it most desirable to have variation between or within blocks? GO TOANSWER 6.

7. Given an RCBD with sampling, what is the cause of sampling error? GO TO ANSWER 7. 8. What precisely, contributes to sampling error sum of squares? GO TO ANSWER 8.

9. Define experimental error for the RCBD. GO TO ANSWER 9.

10. Write the linear additive model for the RCBD, and then define each term. GO TOANSWER 10.

11. What should you conclude if you find that the F-test for replications is significant in anRCBD? GO TO ANSWER 11.

12. Assume you have an RCBD with t=8 and r=3. If F for treatments is significant at the .05level, what can you conclude? What additional statistical procedure would be informative? GOTO ANSWER 12.

13. Assume you have an RCBD in which you have six treatments and four replications. Datafrom two experimental units are missing. GO TO ANSWER 13.

A) List the procedure you would follow to obtain estimates of the missing data.

B) How many degrees of freedom would you have for error?

14. The formula to calculate an estimate for a missing plot for an RCBD is:

(rB + tT - G)/(r - 1)(t - 1). Define each term in the numerator. GO TO ANSWER 14.

15. Assume you have an RCBD and have ten treatments with four reps. If treatment `A` isrepeated three times within each rep., show the degrees of freedom for the sources of variationmaking up experimental error. GO TO ANSWER 15.

16. Assume five treatments and four replications of an RCBD and treatment two is repeatedtwice in every block and treatment five is repeated three times in every block. List all the sources of variation and their degrees of freedom. Show possible randomizations of replicationsone and two. GO TO ANSWER 16.

17. Assume you have 6 treatments and 3 replications in an RCBD. Treatments 1 and 2 repeatedtwo times in each replication. List the randomization procedure and then show how theprocedure is used to do the randomization. GO TO ANSWER 17.

18. A) given the following data, compute treatment sums of squares GO TO ANSWER 18.

T1 | T1 | T2 | T3 | T4 | T5 | Y_{.j} | |

R1 | 36 | 34 | 46 | 38 | 52 | 57 | 263 |

R2 | 42 | 43 | 50 | 47 | 53 | 51 | 286 |

R3 | 45 | 49 | 52 | 44 | 61 | 64 | 315 |

Y_{i.} | 123 | 126 | 148 | 129 | 166 | 172 | 864 |

B) How many degrees of freedom are there for treatments, experimental error, and t1/reps?

C) Calculate the standard error of a difference appropriate for comparing treatments one andfour, assuming the error mean square is 722.

19. Given the following data:

T1 | T2 | T3 | T4 | T5 | T6 | Y_{.j} | |

R1 | 50 | 68 | 28 | 32 | 20 | 17 | 215 |

R2 | 62 | 74 | 36 | 34 | 30 | 29 | 265 |

R3 | 75 | 93 | 49 | 53 | 37 | 40 | 347 |

Y_{i.} | 187 | 235 | 113 | 119 | 87 | 86 | 827 |

A) Compute treatment SS, assuming a normal RCBD.

B) Compute treatment SS, assuming an RCBD with T3=T4 and T5=T6, that is, two treatmentsare each repeated twice in each replication.

C) Compute `T5=T6/reps` SS (same assumption as part b).

D) How many degrees of freedom would you have for the source of variation listed in part C?

E) Continuing the conditions stated in part B, assume experimental error SS equals 55.56,compute the standard error you would need to compare T1 with T3=T4.

20. From the following analysis of variance table, compute the mean square error and anestimate of sigma square epsilon. GO TO ANSWER 20

SOV | df | SS |

Replication | 3 | 237 |

Treatments | 11 | 594 |

ExperimentalError | 33 | 627 |

Sampling Error | 96 | 672 |

21. Assume an RCBD and you calculate the LSD and Duncan`s mrt at the 95% level ofsignificance. If the LSD was 11.5, would the LSR value for p=2 be larger, equal, or smaller? GO TO ANSWER 21.

22. Assume you have an RCBD with six treatments and four replications, but treatment one isrepeated three times within each replication. GO TO ANSWER 22.

A) Show symbolically how to calculate treatment sums of squares. Use actual numbers for thedenominators.

B) Show, in detail, how to calculate an lsd comparing treatment one with treatment 3. Howmany degrees of freedom would be associated with the `t` value?

23. Given an RCBD with 8 treatments and 4 replications, assume you have two missing plots,one in treatment 4 rep 2, and one in treatment 7 rep 4. What are the degrees of freedom fortreatments and for experimental error? GO TO ANSWER 23.

24. Given the following information, list the degrees of freedom for the indicated sources ofvariation. GO TO ANSWER 24.

RCBD, t=12, r=4 | Latin square |

T1 repeated 3 times per rep | r=5, 3 latin squares |

T1 within reps | Square x Treatment |

Experimental Error | Experimental Error |

25. What is the normal range in size for latin square? GO TO ANSWER 25.

26. What is the main disadvantage of a) small numbers of treatments in a latin square and b)large numbers of treatments in a latin square. GO TO ANSWER 26.

27. What is the minimum size latin square on which a complete analysis can be obtained. GOTO ANSWER 27.

28. Why are 3x3 latin squares seldom used? GO TO ANSWER 28.

29. If you were conducting an experiment with 20 treatments, what would be the biggestdisadvantage of using a latin square? GO TO ANSWER 29.

30. Assume a 6x6 latin square. If you have data missing for two experimental units, how manydegrees of freedom do you have for treatments and for error? GO TO ANSWER 30.

31. Diagram three different 4x4 standard latin squares. GO TO ANSWER 31.

32. List the sources of variation and degrees of freedom for the combined analysis of 3, 4x4 latinsquares. GO TO ANSWER 32.

33. Tell (or illustrate) precisely how to compute the treatment SS and treatment x square ss ifyou have two 4x4 latin squares. GO TO ANSWER 33.

34. A latin square design has 50 degrees of freedom for sampling error and 74 degrees offreedom for total. How many degrees of freedom would you have for experimental error? GOTO ANSWER 34.

35. How many degrees of freedom are there for exp. Error in a 7x7 latin square? GO TOANSWER 35.

36. Diagram a standard 4x4 latin square and show how to randomize it. Explain randomizationprocedure. GO TO ANSWER 36.

37. Use the following data to compute 1) SS A), 2) SS B and 3) SS AxB. GO TO ANSWER 37.

R1 | R2 | |

a_{0}b_{0}c_{0} | 7 | 9 |

a_{0}b_{0}c_{1} | 12 | 11 |

a_{0}b_{1}c_{0} | 13 | 16 |

a_{0}b_{1}c_{1} | 18 | 20 |

a_{1}b_{0}c_{0} | 15 | 19 |

a_{1}b_{0}c_{1} | 22 | 20 |

a_{1}b_{1}c_{0} | 24 | 27 |

a_{1}b_{1}c_{1} | 33 | 29 |

38. What is the difference, if any, between a 3 to the fourth power factorial and a 3 x 4 factorial? GO TO ANSWER 38.

39. Assume you have a 2 to the third power factorial, with treatments arranged in a latin square. What would be the size of the latin square? GO TO ANSWER 39.

40. Assume you have a 3 x 4 factorial in a RCBD with 5 blocks. Factor A is fixed and B israndom. Show how to compute the variance of a treatment mean for A, B, and AB. Use actualnumbers for the denominators. GO TO ANSWER 40.

41. Assume you have an RCBD with a 2 to the third power factorial arrangement. GO TOANSWER 41.

A)Write the expected mean square for B if A and B are fixed and C is random. B) What mean square would be the appropriate denominator in making an F-test for B? 42. Assume you apply a herbicide (rates of 4, 8, and 12 ounces per acre) to each of six wheatvarieties using a CRD with r=4, a 3 x 6 factorial arrangement. List all sources of variation anddegrees of freedom. GO TO ANSWER 42.

43. Indicate the degrees of freedom for all error terms given the following conditions (R=4, A=5,B=2, C=3): RCBD GO TO ANSWER 43.

A) Factorial

B) Given situation A, what is the formula for the standard error between two B means, assumingall factors are fixed. Use numbers where possible.

44. Write the expected mean square for C assuming A and C are fixed variables and B is arandom variable. GO TO ANSWER 44.

45. What would be the appropriate denominator to make an F-test of C, given the conditions ofquestion 44? GO TO ANSWER 45.

46. Write the expected mean square for B, if A is random and B and C are fixed. GO TOANSWER 46.

47. Given 3 factors, A, B, and C, write the expected mean square for B if A and B are randomand C is fixed. GO TO ANSWER 47.

48. Write the expected mean square for AxD, assuming a four factor factorial with A and Bfixed, and C and D random. What term would be the denominator of an F-test of the AxDmean square? GO TO ANSWER 48.

49. Given the following table of treatment totals, r=5, compute the main effect of A and theeffect of AxB. GO TO ANSWER 49.

a1 | a2 | |

b1 | 25 | 50 |

b2 | 40 | 80 |

50. From the following table of treatment means, compute the simple effects of A, simple effectsof B, main effect of A, main effect of B, and interaction effect. Label each answer. GO TOANSWER 50.

a1 | a2 | |

b1 | 60 | 75 |

b2 | 70 | 100 |

**PlSc 724 - Answers to Exam 2 Practice Questions**

ANSWER 1. Soil uniformity - if experimental units are uniform, a CRD probably is best. Ifexperimental units are not uniform overall, but blocks of uniform experimental units can be used,an RCBD is best. If there are gradients in two directions, consider a latin square. Yet, a 10x10latin square is quite large. You might lose as much as you gained.

ANSWER 2. Highest - CRD, lowest - latin square.

ANSWER 3. CRD - each experimental unit is randomly assigned.

RCBD - each replicate is grouped into a block and treatments are randomized within eachreplicate.

ANSWER 4. CRD

A | B | C | A | D |

D | C | D | B | E |

E | A | E | B | C |

RCBD

C | A | D | B | E |

A | E | B | C | D |

C | B | E | A | D |

ANSWER 5. To keep the variability among experimental units within a block as small aspossible and to maximize differences among blocks. This design contributes to increasedprecision by detecting treatment differences.

ANSWER 6. Between

ANSWER 7. Sampling error is due to a failure of sample observations to be the same withinexperimental units.

ANSWER 8. SS Sampling Error = SS total - SS among Expt. Unit Total.

ANSWER 9. Experimental error is due to the failure of treatment observations to have the samerelative rank in all replicates.

ANSWER 10. Y_{ij}=µ +_{i}+ _{j} +_{ij}

µ estimated by Y.. bar

_{i} estimated by Y_{i.} bar - Y.. bar

_{j} estimated by Y_{.j} bar - Y.. bar

_{ij} estimated by Y_{ij} - Y_{i.} bar + Y_{.j} bar + Y.. bar

ANSWER 11. That you were successful and correct in not choosing the CRD as the design touse.

ANSWER 12. That some treatments are significantly different at alpha =0.05. To see which treatments are significantly different from each other we could use the LSD or DMRT.

ANSWER 13. a) I. Values are estimated for all values by one using the formula:

__(Y _{i.} bar+ Y_{.j} bar__)/2

ii. The missing plot equation then is used to estimate the final plot.

iii. Using the number calculated in part ii, estimate the other missing value using the formulapresented above.

iv. Keep doing part ii and iii until the estimated values stay the same.

ANSWER 14. r = number of reps

B = block total containing missing plot

t = number of treatments

T = treatment total containing missing plot

SOV | df |

Rep | 3 |

Trt | 9 |

Check(Rep) | 8 |

Error | 27 |

Total | 47 |

SOV | df |

Rep | 3 |

Trt | 4 |

Check2(Rep) | 4 |

Check5(Rep) | 8 |

Error | 12 |

Total | 31 |

Randomization example

Rep 1 T1 T4 T2 T5 T2 T5 T3 T5

ANSWER 17. Treatments to be assigned at random to experimental units in each replicate. Each replicate randomized separately. To randomize, use the random number table and assigncoded treatments to eight experimental units per replicate.

ANSWER 18. a) 756.5

b) 4,8,3

ANSWER 19. a) 6033.61

b) 6027.44

c) 9.50

d) 3

ANSWER 20. Mean square error = 19

Estimate of sigma square epsilon = 4

ANSWER 21. Equal

Y_{ 1.}^{2}/12 + Y_{ 2.}^{2}/4 + ... + Y_{ 6.}^{2}/4 - Y^{2}../32

b) lsd =s _{D bar} x t

where s _{D bar} = Error MS[(12 + 4)/(12 x 4)]

ANSWER 23. Trt df = 7

ANSWER 24. T1(reps) = 8 trt x sq = 8

Exp. Error = 33 Exp. Error = 36

ANSWER 25. 5 x 5 to 8 x 8

ANSWER 26. a) Small df in error term

ANSWER 27. 3 x 3

ANSWER 28. Small df in error term

ANSWER 29. Cost

ANSWER 30. trt = 5

ANSWER 31. 1.

A | B | C | D |

B | C | D | A |

C | D | A | B |

D | A | B | C |

2)

A | B | C | D |

B | A | D | C |

C | D | A | B |

D | C | A | B |

3)

A | B | C | D |

B | D | A | C |

C | A | D | B |

D | C | A | B |

SOV | df |

Square | 2 |

Row(Square) | 9 |

Column(Square) | 9 |

Treatment | 3 |

Square xTreatment | 6 |

Error | 18 |

Total | 47 |

T1 | T2 | T3 | T4 | Y_{.j} | |

Square1 | 10 | 20 | 30 | 40 | 100 |

Square2 | 20 | 30 | 40 | 50 | 140 |

Y_{i.} | 30 | 50 | 70 | 90 | 240 |

Trt SS = (30^{2} + ... + 90^{2} )/8 - 240^{2}/32

Sq x Trt SS = (10^{2} + ... + 50^{2} )/4 - CF - Sq SS - TRT SS

ANSWER 34. 12

ANSWER 35. 30

ANSWER 36. i) Draw standard square

ii) Randomize all columns

iii) Randomize all but first row

ANSWER 37. a) SS A = 430.5625

b) SS B = 264.0625

ANSWER 38. A 3 to the fourth power factorial has four factors each having 3 levels. A 3 x 4factorial has factor A with 3 levels and factor B with 4 levels.

ANSWER 39. 8 x 8

ANSWER 40. Variance of a treatment mean for A = AB MS/(5x4)

Variance of a treatment mean for B = Error MS/(5x3)

Variance of a treatment mean for AxB = Error MS/(5)

ANSWER 41. a) B= ^{2} + ra^{2}_{BC} + rack^{2}_{B}

b) Use estimated mean square for BC. BC = ^{2} + ra^{2} _{BC}.

SOV | df |

Variety | 5 |

Rate | 2 |

Variety xRate | 10 |

Error | 54 |

Total | 71 |

ANSWER 43. a) 87

b) [(2 x Error MS)/(4x5x3)]^{1/2}

ANSWER 44. C MS = ^{2} + ra^{2}_{BC} + rabk^{2}_{C}

ANSWER 45. BC MS

ANSWER 46. B MS = ^{2} + rc^{2}_{AB} + rac k^{2}_{B}

ANSWER 47. B MS = ^{2} + rc^{2}_{AB} + rac^{2}_{B}

ANSWER 48. AD MS =^{2} + rb^{2}_{ACD} + rbc^{2}_{AD}

denominator would be the ACD EMS

ANSWER 49. Main effect A = 6.5

ANSWER 50. Simple effect of A at b1 = 15

Simple effect of A at b2 = 30

Main effect of A = 22.5

Simple effect of B at a1 = 10

Simple effect of B at a2 = 25

Main effect of B = 17.5

Interaction = 7.5