Question 1

The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends of the type of capping material used or variability in different batches. Each number is a cell total

Accepted Answer

The answer of The accompanying data was obtained in an...

Question 2

The strength of concrete used in commercial construction tends to vary from one batch to another. Consequently, small test cylinders of concrete sampled from a batch are "cured" for periods up to about 28 days in temperature- and moisture-controlled environments before strength measurements are made. Concrete is then "bought and sold on the basis of strength test cylinders". The accompanying data resulted from an experiment carried out to compare three different curing methods with respect to compressive strength (MPa). Analyze this data. $$\begin{array} { c c c c } 
\hline \text { Batch } & \text { Method A } & \text { Method B } & \text { Method C } \
\hline 1 & 30.2 & 33.2 & 30.0 \
\hline 2 & 28.6 & 30.1 & 32.1 \
\hline 3 & 29.5 & 31.7 & 30.0 \
\hline 4 & 31.4 & 34.1 & 33.0 \
\hline 5 & 30.0 & 32.5 & 31.9 \
\hline 6 & 26.4 & 28.8 & 27.3 \
\hline 7 & 27.7 & 27.9 & 30.2 \
\hline 8 & 31.9 & 31.9 & 33.1 \
\hline 9 & 26.1 & 29.0 & 28.7 \
\hline 10 & 28.1 & 28.9 & 32.7 \
\hline
\end{array}$$

Accepted Answer

The answer of The strength of concrete used in commercial...

Question 3

A four-factor ANOVA experiment was carried out to investigate the effects of fabric (A), type of exposure (B), level of exposure (C), and fabric direction (D) on extent of color change in exposed fabric as measured by a spectrocolorimeter. Two observations were made for each of the three fabrics, two types, three levels, and two directions, resulting in MSA = 2207.329, MSB = 47.255, MSC = 491.783, MSD = .044, MSAB = 15.303, MSAC = 275.446, MSAD = .470, MSBC = 2.141, MSBD = .280, MSE = .977, and MST = 93.621 ("Accelerated Weathering of Marine Fabrics," J. Testing and Eval.,. 1992: 139-143). Assuming fixed effects for all factors, carry out an analysis of variance using $\alpha = .01$ for all tests and summarize your conclusions.

Accepted Answer

The answer of A four-factor ANOVA experiment was carried out...

Question 4

In an experiment to see whether the amount of coverage of light-blue interior paint depends either on the brand of paint or on the brand of roller used, 1 gallon of each of four brands of paint was applied using each of three brands of roller, resulting in the following data (number of square feet covered). $$\begin{array} { l l | c c c } 
\hline & &  { \text { Roller Brand } } \
\hline & & \mathbf { 1 } & \mathbf { 2 } & \mathbf { 3 } \
\text { Paint Brand } & \mathbf { 2 } & 404 & 396 & 401 \
& \mathbf { 2 } & 396 & 394 & 397 \
& \mathbf { 3 } & 389 & 392 & 394 \
& \mathbf { 4 } & 394 & 387 & 393 \
\hline
\end{array}$$ 
a. Construct the ANOVA table.
b. State and test hypotheses appropriate for deciding whether paint has any effect on coverage. Use $\alpha - .05$ 
c. Repeat part (b) for brand of roller.
d. Use Tukey's method to identify significant differences among brands. Is there one brand that seems clearly preferable to the others?

Accepted Answer

The answer of In an experiment to see whether the...

Question 5

Which of the following statements are not true?

Accepted Answer

The answer of Which of the following statements are not...

Question 6

The accompanying data resulted from a $2^3$ experiment with three replications per combination of treatments designed to study the effects of concentration of detergent (A), concentration of sodium carbonate (B), and concentration of sodium carboxymethyl cellulose (C0 on cleaning ability of a solution in washing tests (a larger number indicates better cleaning ability than a smaller number). $$\begin{array} { c c c c c c } 
\hline \text { Factor Levels }  \
A & B & C & \text { Condition } & \text { Observations } \
\hline 1 & 1 & 1 & ( 1 ) & \begin{array} { c } 
106,93 , \
116
\end{array} \
\hline 2 & 1 & 1 & a & 198,200,214 \
\hline 1 & 2 & 1 & b & 197,202 , \quad 185 \
\hline 2 & 2 & 1 & a b & 329,331,307 \
\hline 1 & 1 & 2 & c & 149 , \quad 169 , \quad 135 \
\hline 2 & 1 & 2 & a c & 243,247,220 \
\hline 1 & 2 & 2 & b c & 255,230,252 \
\hline 2 & 2 & 2 & a b c & 383,360,364 \
\hline
\end{array}$$ 
a. After obtaining cell totals  
compute estimates of $\beta _ { 1 } , \gamma _ { 11 } ^ {  λ c } , \text {and } \gamma^ {  λ c }  _ { \mathbf {210 }  }$ 
b. Use the cell totals along with Yate's method to compute the effect contrasts and sums of squares. Then construct an ANOVA table and test all appropriate hypotheses using $\alpha = .05$

Accepted Answer

The answer of The accompanying data resulted from a $2^3$...

Question 7

Which of the following statements are not true?

Accepted Answer

The answer of Which of the following statements are not...

Question 8

The current (in $\mu A$ ) necessary to produce a certain level of brightness of a television tube was measured for two different types of glass and three different types of phosphor, resulting in the accompanying data: $$\begin{array} { l | c c c } 
\hline & { \text { Phosphor Type } } \
& \mathbf { 1 } & \mathbf { 2 } & \mathbf { 3 } \
\hline \text { Glass 1 } & 280,290,285 & 300,310,295 & 270,285,290 \
\text { Type 2 } & 230,235,240 & 260,240,235 & 220,225,230 \
\hline
\end{array}$$ Assuming that both factors are fixed, test $H _ { o A B } \text { versus } H _ { oA B }$ at level .01. Then if $H _ { o A B }$ cannot be rejected, test the two sets of main effect hypotheses.

Accepted Answer

The answer of The current (in $\mu A$ ) necessary...

Question 9

Which of the following statements are true?

Accepted Answer

The answer of Which of the following statements are true?
...

Question 10

The following equation SST = SSA + SSB + SSC + SSE applies to which ANOVA model?

Accepted Answer

The answer of The following equation SST = SSA +...

Question 11

Which of the following statements are true?

Accepted Answer

The answer of Which of the following statements are true?
...

Question 12

A data from an experiment to assess the effects of vibration (A), temperature cycling (B), altitude cycling (C), and temperature for altitude cycling and firing (D) on thrust duration are shown below. Use the Yates method to obtain sums of squares and the ANOVA table. Then assume that three- and four-factor interactions are absent, pool the corresponding sums of squares to obtain an estimate of $\sigma ^ { 2 }$ and test all appropriated hypotheses at level .05.

Accepted Answer

The answer of A data from an experiment to assess...

Question 13

The accompanying data table gives observations on total acidity of coal samples of three different types, with determinations made using three different concentrations of ethanolic NaOH.
Type of Coal $$\begin{array} { c c | c c c } 
&& { \text { Morwell } } & \text { Yalloum } & \text { Maddingley } \
\hline & .404 \mathrm {~N} & 8.27,8.17 & 8.66,8.61 & 8.14,7.96 \
\text { NaOH Conc. } & .626 \mathrm {~N} & 8.03,8.21 & 8.42,8.58 & 8.02,7.89 \
& .786 \mathrm {~N} & 8.60,8.20 & 8.61,8.76 & 8.13,8.07 \
& & & & \
\hline
\end{array}$$ Additionally, $$\sum _ { i } \sum _ { j } \sum _ { k } x _ { j j k } ^ { 2 } = 1240.1525 \text { and } \sum _ { i } \sum _ { j } X_{ij\square} ^ { 2 } = 2479.9991$$  
a. Assuming both effects to be fixed, construct an ANOVA table, test for the presence of
ANOVA table, test for the presence of interaction, and then test for the presence of main effects for each factor (all using level .01).
b. Use Tukey's procedure to identify significant differences among the types of coal.

Accepted Answer

The answer of The accompanying data table gives observations on...

Question 14

The number of miles useful tread wear (in 1000's) was determined for tires of five different makes of subcompact car (factor A, with I = 5) in combination with each of four different brands of radial tires (factor B, with J = 4), resulting in IJ = 20 observations. The values SSA = 30, SSB = 45, and SSE = 60 were then computed. Assume that an additive model is appropriate.
a. Test $H _ { 0 } : \alpha _ { 1 } = \alpha _ { 2 } = \alpha _ { 3 } = \alpha _ { 4 } = \alpha _ { 5 } = 0$ 
(no differences in true average tire lifetime due to makes of cars) versus $H _ { a } : \text { at least one } \alpha _ { i } \neq 0$ 
using a level .05 test. 
b. $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = 0$ 
(no differences in "true" average tire lifetime due to brands of tires) versus $H _ { a } : \text { at least one } \beta _ { \mathrm { j } } \neq 0$ 
using a level .05 test.

Accepted Answer

The answer of The number of miles useful tread wear...

Question 15

Because of potential variability in aging due to different castings and segments on the castings, a Latin square design with N = 7 was used to investigate the effect of heat treatment on aging. With A = castings, B = segments, C = heat treatments, summary statistics include $x _ { 000 } = 3815.8 , \sum x _ { 00k } ^ { 2 } = 297,155.01$ and $\sum \sum x _ { y( k ) } ^ { 2 } = 297,317.65 \text {. }$ Obtain the ANOVA table and test at level .05 the hypothesis that heat treatment has no effect on aging.

Accepted Answer

The answer of Because of potential variability in aging due...

Question 16

A particular county in Indiana employs three assessors who are responsible for determining the value of residential property in the county. To see whether these assessors differ systemically in their assessments, five houses are selected, and each assessor is asked to determine the market value of each house. With factor A denoting assessors (I = 3) and factor B denoting houses (J=5), suppose SSA = 12, SSB = 110, and SSE = 26.
a. Test $H _ { o } : \alpha _ { 1 } = \alpha _ { 2 } = \alpha _ { 3 } = 0 \text { at 1evel } 05 . H _ { o }$ 
states that there are no systemic differences among assessors). 
b. Explain why a randomized block experiment with only 5 houses was used rather than a one-way ANOVA experiment involving a total of 15 different houses with each assessor asked to assess 5 different houses (a different group of 5 for each assessor).

Accepted Answer

The answer of A particular county in Indiana employs three...

Question 17

The output of a continuous extruding machine that coats steel pipe with plastic was studied as a function of the thermostat temperature profile (A, at three levels), type of plastic (B, at three levels), and the speed of the rotating screw that forces the plastic through a tube-forming die (C, at three levels). There were two replications (L = 2) at each combination of levels of the factors, yielding a total of 54 observations on output. The sums of squares were SSA = 14,144.44, SSB = 5511.27, SSC = 244,696.39, SSAB = 1069.62, SSAC = 62.67, SSBC = 331.67, SSE = 3127.50, and SST = 270,024.33.
a. Construct the ANOVA table.
b. Use appropriate F tests to show that none of the F ratios for two- or three-factor interactions is
at level .05.
c. Which main effects appear significant?
d. With $x _ { 0010 } = 8242 , x _ { 0020 } = 9732 \text {, and } x _ { 0030 } = 11,210$ 
use Tukey's procedure to identify significant
differences among the levels of factor C.

Accepted Answer

The answer of The output of a continuous extruding machine...

Question 18

The following summary quantities were computed from an experiment involving four levels of nitrogen (A), two times of planting (B), and two levels of potassium (C). Only one observation (N content, in percentage, of corn grain) was made for each of the 16 combinations of levels.
SSA = .22625 SSB = .000025 SSC = .0036 SSAB = .004325
SSAC = .00065 SSBC = .000625 SST = .2384.
a. Construct the ANOVA table.
b. Assume that there are no three-way interaction effects, so that MSABC is a valid estimate of $\sigma ^ { 2 }$ 
and test at level .05 for interaction and main effects. 
c. The nitrogen averages are $\bar{x}_{100}=1.1200, \bar{x}_{200}=1.3025, \bar{x}_{300}=1.3875 \text {, and } \bar{x}_{400}=1.4300$ 
Use Tukey's method to examine differences in percentage N among the nitrogen levels $\left( Q _ { .05,4,3 } = 6.82 \right)$

Accepted Answer

The answer of The following summary quantities were computed from...

Question 19

Which of the following statements are not true?

Accepted Answer

Tukey's multiple comparison procedure can be used in three-factor (or more) ANOVA as well as two-factor ANOVA.

Question 20

In an experiment to assess the effects of curing time (factor A) and type of mix (factor B) on the compressive strength of hardened cement cubes, three different curing times were used in combination with four different mixes, with three observations obtained for each of the 12 curing time-mix combinations. The resulting sums of squares were computed to be SSA = 30,763.0, SSB = 34,185.6, SSE = 97,436.8, and SST = 205,966.6.
a. Construct an ANOVA table.
b. Test at level .05 the null hypothesis $H _ { o A B } : \text { all } \gamma _ { \mathrm { ij } } { } ^ { \prime } s = 0$ 
(no interaction of factors) against $H _ { o A B } \text { : at least one } \gamma _ { \mathrm { ij } } \neq 0$ 
c. Test at level .05 the null hypothesis $H _ { o A } : \alpha _ { 1 } = \alpha _ { 2 } = \alpha _ { 3 } = 0$ 
(factor A main effects are absent) against $H _ { o A } \text { : at least one } \alpha _ { i } \neq 0$ 
d. Test $H _ { o B } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = 0 \text { ver sus } H _ { o B } :$ 
at least one $\beta _ { j } \neq 0$ 
using a level .05 test. 
e. The values of the $\bar { x } _ { 1 00 } \text { 's were } \bar { x } _ { 100 } = 4010.88 , \bar { x } _ { 2 00 } = 4029.10 \text {, and } \bar { x } _ { 300 } = 3960.02$ 
Use Tukey's procedure to investigate significant differences among the three curing times.

Accepted Answer

The answer of In an experiment to assess the effects...

Quiz 11: Multifactor Analysis of Variance

The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends of the type of capping material used or variability in different batches. Each number is a cell total

Which of the following statements are not true?

Which of the following statements are not true?

Which of the following statements are true?

The following equation SST = SSA + SSB + SSC + SSE applies to which ANOVA model?

Which of the following statements are true?

Which of the following statements are not true?