stat 423 exam

Stat 423 Section 02 (Robbie) Name ___________________________________ Spring 2020 Exam 2 (90 points) ID Number _________________________ Part I. Workout Problems. Show solution in support of your answers. Unsupported answers will not receive full credit. (51 points) 1. Each of 𝐽 = 2 operators were asked to make 𝐾 = 2 independent measurements on 𝐼 = 10 injection-molded

parts during production. A two-way-interaction ANOVA table for analyzing the data is given below. Responses were 0.001 mm above the nominal value of 685 mm.

Source df SS MS F P-value A = Part 9 11750 1305.5 ? 6.06e-05 B = Operator 1 648 648.0 ? 0.0603 AB Interaction 9 2538 282.0 ? 0.1484 Error 20 3269 163.5 Total 39 18205

a. Operator effect is fixed. Let 𝛽! be the main effect of Operator 𝑗. Complete the hypothesis test below. [12 pts]

i. 𝐻": _________________________________________________ versus 𝐻#: {𝐻" is false.}

ii. 𝐹 test statistic = ____________________ iii. Rejection Region (𝛼 = 0.05) = _______________________ P-value = __________________ iv. Circle one: Reject Retain

b. A plot of mean is given on the right. Does this plot agree with the ANOVA results regarding the interactions between Parts and Operators? Explain briefly. [5 pts]

c. Assume that Part effects are fixed. Compute 𝑤 in the T Method (𝛼 = 0.05) for comparing levels of Part and use it to determine if Part 2 (�̅�$. = −327.75) and Part 3 (�̅�&. = −318.50) are statistically different. [8 pts]

-3 30

-3 10

-2 90

-2 70

Part

A ve

ra ge

R es

po ns

e

1 10 2 3 4 5 6 7 8 9

Operator

2 1

Problem 1 (continued). d. The ANOVA is revised to combine {AB Interaction} and

Error. See above. The effect of A=Part is random because parts were randomly selected from a large batch, while B=Operator effect is considered fixed. Estimate the percent of total variation that is due to the differences between parts. [10 pts]

2. A full 23 factorial experiment was conducted to study the effect of A={Flow Rate}, B={Rotational Speed}, and

C={Type of Mud} on the advance rate of a small stone drill. Consider the full model 𝑋'!( = 𝜇 + 𝛼' + 𝛽! + 𝜒( + all interactions + 𝜀'!(

where 𝛼' is the main effect of A, 𝛽! is the main effect of B, and 𝜒( is the main effect of C. An ANOVA from fitting this model is given below. The colon “:” in the table denotes factor interaction.

Source Df SS MS F P-value A 1 134.6 134.6 90.267 1.24e-05 B 1 533.1 533.1 357.536 6.33e-08 C 1 42.7 42.7 28.603 0.000687 A:B 1 1.0 1.0 0.676 0.434895 A:C 1 0.1 0.1 0.060 0.812884 B:C 1 3.9 3.9 2.585 0.146558 A:B:C 1 0.5 0.5 0.320 0.587395 Error 8 11.9 1.5 Total 15 727.8

a. Use the ANOVA results to compute the missing 𝑅$ and 𝑅adj $ for the models below. [12 pts]

Model 𝑹𝟐 𝑹𝒂𝒅𝒋 𝟐

𝑋'!( = 𝜇 + 𝛼' + 𝛽! + 𝜒(+ all interactions + 𝜀'!( 98.36% 96.93%

𝑋'!( = 𝜇 + 𝛼' + 𝛽! + 𝜒( + 𝛾'! 01 + 𝜀'!(

?

96.94%

𝑋'!( = 𝜇 + 𝛼' + 𝛽! + 𝜒( + 𝜀'!( 97.62% 97.02%

𝑋! = 𝛽! + 𝜀!

73.26%

?

b. Based on the results in the ANOVA table and in (a), what small but adequate (parsimonious) model would you

recommend to describe the behavior of the advance rate of the stone drill? Explain briefly. [4 pts]

Part II. Multiple Choice. Circle the letter of the correct/best answer. (39 pts) Problems 1-4: Below is Yates algorithm applied to data from a full 23 factorial with factors A, B and C.

Treatment Means Cycle 1 Cycle 2 Cycle 3 Fitted Effect (1) 6.23 14.11 40.82 127.82 15.98 a 7.88 26.71 87.00 5.20 0.65 b 12.98 38.20 2.39 23.20 2.90 ab 13.73 48.80 2.81 � ? c 18.28 1.64 12.61 46.18 5.77 ac 19.92 0.75 10.60 0.41 0.05 bc � 1.64 -0.89 -2.01 -0.25 abc 24.98 1.16 -0.48 0.41 ?

The model fitted here is 𝑋'!( = 𝜇 + 𝛼' + 𝛽! + 𝜒( + 𝛾'!

01 + 𝛾'( 02 + ⋯+ 𝜖'!( where ai = main effect of A at level i,

bj = main effect of B at level j, ck = main effect of C at level k, and g’s are interactions between factors.

1. Compute the value labeled �. A. 23.34 B. 23.82 C. 26.14 D. 26.62

2. Compute the value labeled �. A. −1.37 B. −0.48 C. 0.41 D. 1.37

3. What is an estimate of 𝛾333012?

A. −0.41 B. −0.05 C. 0.05 D. 0.41

4. A Normal probability plot of the fitted effects (excluding 15.98) is given on the right. Which effect (main effect or interaction) is most significant according to this plot? A. Main effect of A B. Main effect of B C. Main effect of C D. Interaction between B and C

0 1 2 3 4 5 6

-1 .0

-0 .5

0. 0

0. 5

1. 0

Normal Q-Q Plot

Fitted Effect

T he

or et

ic al

Q ua

nt ile

s

5. In a two-factor experiment, factor A has 4 levels, factor B has 3 levels, and there is only one observation for each of the 12 treatments (factor combinations). Which of the following statements is FALSE? A. We cannot perform an F test for interactions between A and B. B. SST has 12 degrees of freedom. C. SSB has 2 degrees of freedom. D. SSA has 3 degrees of freedom.

6. Which of the following is TRUE about a Latin Square design (for 3 factors)? A. Factors A, B and C can have different number of levels. B. We must assume that there are no 2 and 3-factor interactions. C. A Latin Square design can be costly to implement because we have to run all factor combinations. D. We cannot apply the T Method on data from a Latin Square design.

7. A primary interest of designing a randomized block experiment is:

A. to produce experimental units that are identical B. to increase the between-treatments variation to more easily detect differences among the treatment

means C. to provide better comparisons of treatments by accounting for differences in experimental units D. (A) and (C)

8. Which of the following statements are true? A. Sums of squares of effects that are judged unimportant can combined to obtain a new error sum of

squares that can be used to perform F test statistics. B. It is possible that 𝑅adj

$ of a smaller model is larger (better) than 𝑅adj $ for a bigger model.

C. The coefficient of determination 𝑅$ for a bigger model is always at least as big as the 𝑅$ of a smaller model.

D. All of the above statements are true. For Problems 9 and 10: In a three-factor experiment suppose that factor A has 3 levels, factor B has 2 levels, and factor C has 3 levels. Also, for each of the combination of levels of the factors, 3 observations were measured. The researcher fitted the full model with main effects, all two-factor interactions, and all three-factor interactions. 9. How many possible treatments are there?

A. 8 B. 11 C. 18 D. 54

10. If all treatments were applied, what are the degrees of freedom (df) for Error in the ANOVA table?

A. 16 B. 22 C. 36 D. 108

Problems 11-13: An experiment studied the effects of A={Hole Size} and B={Distance of Hole from Edge} on tensile strength of holes drilled on aluminum strips. An ANOVA table for analyzing the data is given below.

Source df SS MS A=Size 2 ? ? B=Distance 1 ? 7.157 AB Interaction 2 0.224 0.112 Error 12 13.651 1.138 Total 17 28.624

11. How many hole sizes were used in the experiment? A. 2 B. 3 C. 6 D. 18

12. What is the sum the of squares for factor A={Hole Size}?

A. 1.138 B. 3.796 C. 7.157 D. 7.592

13. A plot of means is given on the below. What conclusion(s) can you derive from this plot? A. Increasing the hole size (A), reduces tensile strength. B. Increasing the hole size (A), increases tensile strength. C. Increasing the distance (B) of the hole from the edge,

increases tensile strength. D. Both (A) and (C).

54 .5

55 .0

55 .5

56 .0

56 .5

57 .0

A=Size

A ve

ra ge

S tre

ng th

0.149in 0.185in 0.221in

B=Distance

1.0in 0.5in