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Chapter 11: Problem 50

The article "Towards Improving the Properties of Plaster Moulds and Castings" (J. Engrg. Manuf., 1991: 265-269) describes several ANOVAs carried out to study how the amount of carbon fiber and sand additions affect various characteristics of the molding process. Here we give data on casting hardness and on wet-mold strength. $$ \begin{array}{llll} \hline \begin{array}{l} \text { Sand } \\ \begin{array}{l} \text { Addition } \\ (\%) \end{array} & \begin{array}{l} \text { Carbon } \\ \text { Fiber } \\ \text { Addition } \\ (\%) \end{array} & & \begin{array}{l} \text { Casting } \\ \text { Hardness } \end{array} & \begin{array}{l} \text { Wet- } \\ \text { Mold } \\ \text { Strength } \end{array} \\ \hline 0 & 0 & 61.0 & 34.0 \\ 0 & 0 & 63.0 & 16.0 \\ 15 & 0 & 67.0 & 36.0 \\ 15 & 0 & 69.0 & 19.0 \\ 30 & 0 & 65.0 & 28.0 \\ 30 & 0 & 74.0 & 17.0 \\ 0 & .25 & 69.0 & 49.0 \\ 0 & .25 & 69.0 & 48.0 \\ 15 & .25 & 69.0 & 43.0 \\ 15 & .25 & 74.0 & 29.0 \\ 30 & .25 & 74.0 & 31.0 \\ 30 & .25 & 72.0 & 24.0 \\ 0 & .50 & 67.0 & 55.0 \\ 0 & .50 & 69.0 & 60.0 \\ 15 & .50 & 69.0 & 45.0 \\ 15 & .50 & 74.0 & 43.0 \\ 30 & .50 & 74.0 & 22.0 \\ 30 & .50 & 74.0 & 48.0 \end{array} \end{array} $$ a. An ANOVA for wet-mold strength gives SSSand \(=705\), SSFiber \(=1278\), SSE \(=843\), and SST \(=3105 .\) Test for the presence of any effects using \(\alpha=.05\). b. Carry out an ANOVA on the casting hardness observations using \(\alpha=.05\). c. Make an interaction plot with sand percentage on the horizontal axis, and discuss the results of part (b) in terms of what the plot shows.

Short Answer

Expert verified
Conduct ANOVA for both wet-mold strength and casting hardness to determine significant effects, then use a plot to explore interactions between sand and fiber on hardness.

Step by step solution

01

Understanding ANOVA Components

For part (a), we are given the components of the ANOVA for wet-mold strength: SSSand (the sum of squares for sand addition) = 705, SSFiber (the sum of squares for carbon fiber addition) = 1278, SSE (the error sum of squares) = 843, and SST (the total sum of squares) = 3105. These components will allow us to perform ANOVA and test for significance of effects.
02

Calculate Mean Squares for Wet-Mold Strength

To perform ANOVA, calculate the mean squares for sand, fiber, and error. Mean square is computed as sum of squares divided by their respective degrees of freedom. We're not given degrees of freedom, so assume typical ANOVA practice for a 2x3 factorial design. MS (mean square) is: - MSSand = SSSand / dfSand - MSFiber = SSFiber / dfFiber - MSError = SSE / dfError. Assume appropriate degrees of freedom for each source based on the experimental setup.
03

Conduct F-tests for Wet-Mold Strength

For the F-test, compare the mean squares against the error mean square to get the F-ratios: - FSand = MSSand / MSError - FFiber = MSFiber / MSError Using a significance level of \(\alpha = 0.05\), consult F-distribution tables to find the critical values for these comparisons. If the calculated F is larger than the critical F, reject the null hypothesis.
04

Interpret ANOVA Results for Wet-Mold Strength

If FSand or FFiber is significant, it indicates that respective variable significantly affects wet-mold strength. State conclusions based on whether any null hypothesis was rejected in the F-test and the corresponding effect's significance.
05

ANOVA Setup for Casting Hardness

Part (b) requires performing a separate ANOVA for casting hardness. Start by calculating sums of squares due to each factor and interaction. Then calculate their respective mean squares utilizing the experimental degrees of freedom and proceed to F-testing as done before.
06

Interpretation of ANOVA for Casting Hardness

Analyze F-tests from hardness results to check for any significant effects. Conclude whether sand addition, fiber addition, or their interaction significantly affects casting hardness.
07

Plot Interaction Effect

For part (c), plot an interaction graph of casting hardness against sand percentage. On this plot, use separate lines for each fiber content percentage. Examine interaction effects: if lines are parallel, there's likely no interaction; if lines intersect, interaction might be significant.
08

Discuss Interaction Plot

Using the plot from Step 7, interpret whether different percentage additions interact uniquely. If interaction is significant, lines will not be parallel. Include observations from the ANOVA results to correlate on the plot.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Analysis
Statistical analysis is the process used to understand data better, identify patterns, and make predictions. It involves various techniques and methods, among which the Analysis of Variance (ANOVA) is quite prominent. ANOVA is especially useful when you want to compare three or more group means to see if at least one is different from the others.
  • In the given exercise, ANOVA is used to assess how different amounts of sand and carbon fiber impact properties like casting hardness and wet-mold strength.
  • This requires calculating sums of squares, which quantify the amount of variation in your data.
  • The sums of squares are split into parts attributed to different sources, such as the main effects (like sand addition) and error (the part not explained by your studied factors).
By examining these, ANOVA helps to determine whether the variation between group means is statistically significant, meaning that it's unlikely to have occurred by random chance.
Understanding these processes can significantly aid in making informed conclusions from experimental data.
Experimental Design
Experimental design is the framework that allows scientists to test hypotheses and interpret their results. It includes planning how to collect, analyze, and interpret the data most effectively.
In the context of this exercise, a factorial design seems to be used, where two factors鈥攕and and carbon fiber鈥攁re varied to explore their individual and combined effects on casting hardness and wet-mold strength.
  • Factorial designs help in understanding interaction effects between factors, which will be discussed later.
  • A typical approach involves setting up groups where the factors vary within expected ranges, such as several percentages of sand and fiber.
These experiments are crucial as they help in establishing cause-and-effect relationships and optimizing product or process conditions.
Good experimental design minimizes biases and variabilities in results, ensuring that observed effects can be confidently attributed to the factors being studied.
Interaction Effects
Interaction effects occur when the impact of one factor on the outcome variable depends on the level of another factor. It's a crucial aspect of multifactorial experiments because it can significantly influence the results.
Consider the case where sand percentage and carbon fiber content influence casting hardness. If changing sand levels only significantly alter hardness at certain fiber levels, it demonstrates interaction.
  • An interaction plot can visualise these effects, typically by graphing one factor on the x-axis and having separate lines for the other factor levels.
  • If the lines intersect or aren't parallel, it indicates interaction between the factors.
Recognizing interaction effects is essential because it may highlight synergies between components affecting the overall result, which can be leveraged for enhanced performance or outcomes in practical applications.
Significance Testing
Significance testing helps decide if the observed effects in your data are real or just due to random chance. By employing statistical tests such as the F-test used in ANOVA, researchers can infer the probability that the results are significant.
  • Each factor's effect is assessed by comparing a calculated statistic (F-ratio) against a critical value from the F-distribution, determined by your level of significance (usually 0.05).
  • If the F-ratio exceeds this critical value, the effect is statistically significant, rejecting the null hypothesis (that there's no effect).
In the exercise, both sand and carbon fiber's effects on wet-mold strength and casting hardness are tested for significance.
This process is essential to validate any findings, ensuring the observed differences are genuine and can have meaningful impacts when applied in real-world settings.

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Most popular questions from this chapter

The accompanying data resulted from an experiment to investigate whether yield from a chemical process depended either on the formulation of a particular input or on mixer speed. $$ \begin{array}{cc|ccc} & & {\text { Speed }} \\ { 3 - 5 } & & \mathbf{6 0} & \mathbf{7 0} & \mathbf{8 0} \\ \hline \text { Formulation } & & 189.7 & 185.1 & 189.0 \\ & & 188.6 & 179.4 & 193.0 \\ & & 190.1 & 177.3 & 191.1 \\ & & 165.1 & 161.7 & 163.3 \\ & & 165.9 & 159.8 & 166.6 \\ & & 167.6 & 161.6 & 170.3 \\ \hline \end{array} $$ A statistical computer package gave \(\mathrm{SS}(\mathrm{Form})=\) \(2253.44, \mathrm{SS}(\) Speed \()=230.81, \mathrm{SS}(\) Form* Speed \()\) \(=18.58\), and SSE \(=71.87\). a. Does there appear to be interaction between the factors? b. Does yield appear to depend on either formulation or speed? c. Calculate estimates of the main effects. d. Verify that the residuals are \(0.23,-0.87,0.63\), \(4.50,-1.20,-3.30,-2.03,1.97,0.07,-1.10\), \(-0.30,1.40,0.67,-1.23,0.57,-3.43,-0.13\), \(3.57\). e. Construct a normal plot from the residuals given in part (d). Do the \(\varepsilon_{i j k}\) 's appear to be normally distributed? f. Plot the residuals against the predicted values (cell means) to see if the population variance appears reasonably constant.

In an experiment to see whether the amount of coverage of light-blue interior latex 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). $$ \begin{array}{lc|ccc} & &{\text { Roller Brand }} \\ & & \mathbf{1} & \mathbf{2} & \mathbf{3} \\ & \mathbf{1} & 454 & 446 & 451 \\ \text { Paint } & \mathbf{2} & 446 & 444 & 447 \\ \text { Brand } & \mathbf{3} & 439 & 442 & 444 \\ & \mathbf{4} & 444 & 437 & 443 \\ \hline \end{array} $$ a. Construct the ANOVA table. [Hint: The com putations can be expedited by subtracting 400 (or any other convenient number) from each observation. This does not affect the final results.] b. State and test hypotheses appropriate for deciding whether paint brand has any effec on coverage. Use \(\alpha=.05\). c. Repeat part (b) for brand of roller. d. Use Tukey's method to identify significan differences among brands. Is there one bran that seems clearly preferable to the others? e. Check the normality and constant variance assumptions graphically.

When both factors are random in a two-way ANOVA experiment with \(K\) replications per combination of factor levels, the expected mean squares are \(E(\mathrm{MSE})=\sigma^{2}, E(\mathrm{MSA})=\sigma^{2}+\) \(K \sigma_{G}^{2}+J K \sigma_{A}^{2}, E(\mathrm{MSB})=\sigma^{2}+K \sigma_{G}^{2}+I K \sigma_{B}^{2}\), and \(E(\mathrm{MSAB})=\sigma^{2}+K \sigma_{G}^{2}\) a. What \(F\) ratio is appropriate for testing \(H_{0 G}: \sigma_{G}^{2}=0\) versus \(H_{\mathrm{a} G}: \sigma_{G}^{2}>0\) ? b. Answer part (a) for testing \(H_{0 A}: \sigma_{A}^{2}=0\) versus \(H_{\mathrm{aA}}: \sigma_{A}^{2}>0\) and \(H_{0 B}: \sigma_{B}^{2}=0\) versus \(H_{\mathrm{a} B}: \sigma_{B}^{2}>0\)

In an experiment to assess the effect of the angle of pull on the force required to cause separation in electrical connectors, four different angles (factor \(A\) ) were used and each of a sample of five connectors (factor \(B\) ) was pulled once at each angle ("A Mixed Model Factorial Experiment in Testing Electrical Connectors," Indust. Qual. Control, 1960: 12-16). The data appears in the accompanying table. $$ \begin{array}{cc|ccccc} & & {\boldsymbol{B}} \\ & & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\ \hline & \mathbf{0}^{\circ} & 45.3 & 42.2 & 39.6 & 36.8 & 45.8 \\ \mathbf{A} & \mathbf{2}^{\circ} & 44.1 & 44.1 & 38.4 & 38.0 & 47.2 \\ & \mathbf{4}^{\circ} & 42.7 & 42.7 & 42.6 & 42.2 & 48.9 \\ & \mathbf{6}^{\circ} & 43.5 & 45.8 & 47.9 & 37.9 & 56.4 \\ \hline \end{array} $$ Does the data suggest that true average separation force is affected by the angle of pull? State and test the appropriate hypotheses at level \(.01\) by first constructing an ANOVA table (SST \(=396.13, \mathrm{SSA}=58.16\), and \(\mathrm{SSB}=246.97\) ).

The following data refers to yield of tomatoes \((\mathrm{kg} /\) plot) for four different levels of salinity; salinity level here refers to electrical conductivity (EC), where the chosen levels were \(\mathrm{EC}=1.6,3.8,6.0\), and \(10.2 \mathrm{nmhos} / \mathrm{cm}\) : \(\begin{array}{rrrrrr}1.6: & 59.5 & 53.3 & 56.8 & 63.1 & 58.7 \\ 3.8: & 55.2 & 59.1 & 52.8 & 54.5 & \\ 6.0: & 51.7 & 48.8 & 53.9 & 49.0 & \\ 10.2: & 44.6 & 48.5 & 41.0 & 47.3 & 46.1\end{array}\) Use the \(F\) test at level \(\alpha=.05\) to test for any differences in true average yield due to the different salinity levels.
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