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Question 1: the SAS code is:
First, we use normal probability plot to investigate the significant effects:
By looking to the normal plot and half-normal plot, we can see Temperature is the only variable that is
much different from the normal line comparing with the other effects.
Moreover, it is clear that Temperature is the only effect that is highly significant among the other
factors. Can see that in both the main effects plot and the lenth plot.
Second, we use analysis of variance to verify our result:
We create the hypothesis test to see which factor does impact Yield:
0 : .
1 : .
Using ANOVA table, we find that Temperature has p-value is 0.0162 with f-value 15.97.
Now by comparing these results with the significance level for 0.05,1,56 = 4.01 = 0.05
so these result confirm what we have found in the first part. Hence we reject the null
hypothesis and conclude that Temperature does impact Yield.
0 : .
1 : .
Using ANOVA table, we find that concentration has p-value is 0.4340 with f-value 0.75.
Now by comparing these results with the significance level for 0.05,1,56 = 4.01 = 0.05.
Hence we do not reject the null hypothesis and conclude that concentration does not
impact Yield.
0 : .
1 : .
Using ANOVA table, we find that catalyst has p-value is 0.8073 with f-value 0.07. Now by
comparing these results with the significance level for 0.05,1,56 = 4.01 = 0.05. Hence
we do not reject the null hypothesis and conclude that catalyst does not impact Yield.
By looking to the residual plots above, we indicate that there may be some problem with
inequality of variance. The normality and linearity look pretty good in other graphs of
Residual.
Question 2
We have unreplicated 23 factorial design, which mean the design have 8 treatment
combinations. The risk of conducting an expereiment with unreplicated design is the model
may contains a lot of niose especially when the dependent variable (yield) is highly
variable. Thus misleading conclusions may result from the experiment. Another problem
with unreplicated design, there’s no estimate of pure error, therefore we need to assum
that certain high-order intercations are negligible, but when the high order intercations are
important we need to investigate the normal probability plot of the estimates of the effects.
The effects that are negligible are normally distributed, and tend to fall along blue line,
whereas significant effects are not normally distributed and will have nonzero means and
will not lie along the blue line. The important effects are the main effect of temperature and
the interaction between temperature and catalyst.
To verify the significant terms using the analysis of variance, we will combine the negligible
effects as an estimate of error. Because the effect of the intercation between temperature
and catalyst is significant, therefore we need to include the main effect of catalyst in the
analysis of variance.
Source
Nparm DF Sum of Squares F Ratio Prob > F
temp
1 1
1058.0000 36.8000 0.0037*
catalyst
1 1
4.5000 0.1565 0.7126
temp*catalyst
1 1
200.0000 6.9565 0.0577
Term
Estimate Std Error t Ratio Prob>|t|
Intercept
66.75 1.895719 35.21 <.0001* temp 11.5 1.895719 6.07 0.0037* catalyst 0.75 1.895719 0.40 0.7126 temp*catalyst 5 1.895719 2.64 0.0577 Based on the table 1, the intercation between temperture and catalyst is significant (F=6.96, p=0.058) at level 0.1 (N.B. main effects are not important when they are involved in significant interactions). The the intercation between temperature and catalyst indicates that catalyst has little effect at low temperature (-1) but a large effect at high temperature (+1). From the table 2, the estimated yield is given by ̂ = 11.5 + 11.5temp + 0.75catalyst + 5temp ∗ catalyst Based on the Normal Quantile Plot (fig 3) the normality assumption seems to be fulfilled and from predicted versus residual plot (fig 4) there’s no pattern, thus the assumption of homoscedasticity is not fulfilled. Question 3 A. Dependent variable distribution Based on the Boxplot below, the distribution of the dependent variable (amount of the metal recovered from an ore given a sample of weight ) is approximately symmetric, without outliers. The mean of the dependent variable is 79.125 with a standard deviation of 10.69. B. Full Factorial ANOVA There are two replications, therefore there is an internal estimate of pure error of high order intercations. From the ANOVA table the model is statistically significant (F=8.85, p=0.003), which means there are at least one effect have a significant effect on the DV. The model explain 88.56% of the total variability containing in DV. In the last 2 tables below summarizes the sums of squares and the percentage contribution of each model term relative to the total sum of squares. Only the main effects of concentration (F=11.11, p=0.01) and time (F=45.31, p<0.001) are significant at level 5%. The main effect of time really dominates this process, accounting for over 73.9% of the total variability explained by the model, whereas the main effect of concentration accounts for about 17.9%. Terms temp conc temp*conc time temp*time conc*time temp*conc*time Contribution 0.4% 17.9% 1.3% 73.9% 0.8% 0.8% 4.8% C. ANOVA for Selected Factorial Model From the previous step we found that only the main effects of time and concentartion have significant effects on DV at level 5%, thus in stage we will estimate the model with only the important effects. The model is significant (F=28.39, p<0.001) which the means different. The model explain 81.4% of the total variation. On the other hand, the main effects of time and concentration are significant at level 5% were (F=11.09, p=.005) and (F=45.7, p<.001) respectively. From the profile plots the estimated average of DV increase significantly when the levels of concentration and time increase from low-level (-1) to high level (+1). D. Assumptions of the selected Factorial Model From the QQ plot, the assumption of the normality seems to be fulfilled given that the quantiles tend to fall along the line except one observation (an outlier). The homoscedasticity seems to be also accepted since there’s no pattern between predicted values and Rstudent (student residuals) but there’s a clear outlier with Rstudent > 3 (1st
observation where y=80 and all factors are in low level -1).
Question 4
A. Dependent variable distribution
Based on the Boxplot the distribution of the DV does not conatins extreme values and slightly
right skewed. The average of DV is 198.33 with standard deviation 128.24, which there’s a
large variability.
B. Main effects ANOVA
There are two nominal independent variables (factors). One independent variable has five
levels (material), and the other has three levels (machines). There are a total of 15
observations who are divided into 15 cells. Therefore each cell contains one observation,
which means we have unreplicated factorial design. Then there’s no estimate of pure error,
therefore we need to assum that high-order intercation is negligible to perform the ANOVA.
1) Model
From the ANOVA table the model is statistically significant (F=41.65, p<.001), which means at least one factor have a significant effect on the DV. The model explain 96.7% of the total variability containing in DV. In the last table below summarize the sums of squares and F- test of each model term. Only the main effects of material (F=60.63, p<.001) is significant at level 5%, whereas the main effect of machines is significant (F=3.71 p=0.073) at level 10%. 2) Model assumption Based on the QQ plot and predicted values versus student residuals plot both assumption seems to be fullfiled, except that there is an extreme value (outlier) with Student > 6!!! In
such sitution we need to invistigate why this observation it is an outlier or we can
transform the DV (e.g. logarithm transformation).
C. Main effect ANOVA with transformed DV
In our case we will transform the DV to logarithm, therefore the DV is defined as
( )
1) Model
After the transformation, F, and R square increases to 57.25 and 97.7% respectively and
both main effects become significant at level 5%.
Based on the profile plot of the factor material the 1st level have the highest estimated
average and 5th material level have a lowest estimated average. The estimated average
in the 2nd and 3rd levels are not statistically different at level 5%.
Based on the profile plot of the factor machines, the 1st level has the lowest estimated
average followed by 2nd level and then 3rd level. However there are no significant
differences between 1st vs 2nd levels and between 2nd vs 3rd levels, but there is a significant
difference between 1st and 3rd levels
2) Model assumption
Both assumptions are clearly valid after the transformation of the DV to logarithm scale.
Question 4
The experiment design conatin two factors (clay and mold). Both factors have three levels.
There are a total of 45 observations who are divided into 9 cells. The design is balanced,
therefore each cell contains five observations, which means we have replicated factorial
design. Then we can include the high-order intercation (aka. Full Factorial ANOVA).
A. Full Factorial ANOVA
1) Model
From the ANOVA table the model is statistically significant (F=13.68, p<.001), which means at least one effect is significant. The model explain 75.25% of the total variability containing in DV. The interaction is significant (F=2.99, p=0.032) at level 5%, which means the effect of one independent variable (clay or mold) on the dependent variable is different at different levels of a second independent variable (clay or mold). On the other hand, both main effects are significant at level 5%. Howevere the main effects are not important when intercation is significant. 2) Model assumptions Based on the QQ plot the quantiles do not lie along the line, thus the normality assumption is not fulfilled. On the other hand, the values of student residuals increase when the predicted values increase (variance increase not constant), thus the homoscedasticity is not fulfilled. To fixe these issues we will transform the DV. B. Full Factorial ANOVA with transformed DV 1) Model After trying many transformations we found that the assumptions become valid when we used the following transformation −3 2 The values of F, and R square increases to 43.44 and 90.6% respectively and the interaction term become not significant (F=2.21, p=.087) at level 5%. However, the F values of the main effects increase significantly from 27.2 to 99.05 for the factor clay and from 21.54 to 70.28 for the factor mold, which means the importance of the main effects increase and the importance of the interaction become negligible. Therefore we need to remove the interaction term from the model. 2) Model assumptions The assumptions are clearly valid without any outlier given that the absolute values of student residuals < 3. C. Main Effects ANOVA with transformed DV 1) Model After removing the intercation term the values of F, and R square increase and decrease to 75.5 and 88.3% respectively. The F values of the main effects increase significantly from 99.05 to 88.32 for the factor clay and from 70.28 to 62.67 for the factor mold, which means the importance of the main effects increase without the interaction term. Based on the profile plot of the factor clay, the 1st level has the highest estimated average and 2nd and 3rd levels have the lowest estimated averages and they are not significantly different at level 5%. Based on the profile plot of the factor mold, the 1st and 2nd levels have the lowest estimated averages (they are significantly different) and 3rd level have the highest estimated average. 2) Model assumptions ... Purchase answer to see full attachment

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