91Ó°ÊÓ

Four different coatings are being considered for corrosion protection of metal pipe. The pipe will be buried in three different types of soil. To investigate whether the amount of corrosion depends either on the coating or on the type of soil, 12 pieces of pipe are selected. Each piece is coated with one of the four coatings and buried in one of the three types of soil for a fixed time, after which the amount of corrosion (depth of maximum pits, in \(.0001\) in.) is determined. The depths are shown in this table: $$ \begin{aligned} &\text { Soil Type (B) }\\\ &\begin{array}{l|lll} & \mathbf{1} & \mathbf{2} & \mathbf{3} \\ \hline \mathbf{1} & 64 & 49 & 50 \\ \mathbf{2} & 53 & 51 & 48 \\ \mathbf{3} & 47 & 45 & 50 \\ \mathbf{4} & 51 & 43 & 52 \\ \hline \end{array} \end{aligned} $$ a. Assuming the validity of the additive model, carry out the ANOVA analysis using an ANOVA table to see whether the amount of corrosion depends on either the type of coating used or the type of soil. Use \(\alpha=.05\). b. Compute \(\hat{\mu}, \hat{\alpha}_{1}, \hat{\alpha}_{2}, \hat{\alpha}_{3}, \hat{\alpha}_{4}, \hat{\beta}_{1}, \hat{\beta}_{2}\), and \(\hat{\beta}_{3}\)

Step by step solution

01

Calculate the Overall Mean

First, calculate the overall mean of all the corrosion data. Sum up all the corrosion depth values and divide by the number of observations. Here, we have 12 observations in total.\[\text{Overall Mean} = \frac{64 + 49 + 50 + 53 + 51 + 48 + 47 + 45 + 50 + 51 + 43 + 52}{12}\]After calculation:\[\text{Overall Mean} \approx 50.25\]

02

Calculate the Treatment Means

Compute the mean corrosion depth for each coating (treatment) and each soil type (block). This involves finding the average for each row (coating) and column (soil type).**Coating Means:**1.\(\mu_1 = \frac{64 + 49 + 50}{3} = 54.33\)2. \(\mu_2 = \frac{53 + 51 + 48}{3} = 50.67\)3. \(\mu_3 = \frac{47 + 45 + 50}{3} = 47.33\)4. \(\mu_4 = \frac{51 + 43 + 52}{3} = 48.67\)**Soil Type Means:**1. \(\mu_1 = \frac{64 + 53 + 47 + 51}{4} = 53.75\)2. \(\mu_2 = \frac{49 + 51 + 45 + 43}{4} = 47\)3. \(\mu_3 = \frac{50 + 48 + 50 + 52}{4} = 50\)

03

Compute Sums of Squares

Calculate the Total Sum of Squares (SST), Treatment Sum of Squares (SSTR), Soil Sum of Squares (SSBS), and Error Sum of Squares (SSE) by using their respective formulas based on the deviation from the means calculated previously. **SST**: Sum of deviations of each observation from the overall mean. **SSTR (Coatings)**: Sum of squared deviations of treatment means from the overall mean. **SSBS (Soil Types)**: Sum of squared deviations of block means from the overall mean. **SSE**: SST - SSTR - SSBS.

04

Fill the ANOVA Table

Populate the ANOVA table with computed sums of squares and mean squares for each source of variation.1. Degrees of Freedom (DF): - DFTotal = 11 - DFCoating = 3 - DFSOILType = 2 - DFError = 62. Calculate Mean Squares (MS) by dividing each SS by its DF.3. Use MS to compute F-statistics: - \(F_{Coating} = \frac{MS_{Coating}}{MS_{Error}}\) - \(F_{Soil} = \frac{MS_{Soil}}{MS_{Error}}\)Compare computed F values to critical value from F-distribution tables at \(\alpha = 0.05\).

05

Interpret the Results

Based on the F-statistics calculated and the critical values, determine if we reject or fail to reject the null hypotheses.- Null hypothesis for coating: All coating effects are equal.- Null hypothesis for soil type: All soil type effects are equal.If the calculated F is greater than the critical F at \(\alpha = 0.05\), reject the null hypothesis.

06

Estimate Effects

Calculate the estimates for the main effects using:\[ \hat{\alpha}_i = \bar{x}_{i\cdot} - \bar{x} \]\[ \hat{\beta}_j = \bar{x}_{\cdot j} - \bar{x} \]For coatings (i) effects:- \(\hat{\alpha}_1 = 54.33 - 50.25\)- \(\hat{\alpha}_2 = 50.67 - 50.25\)- \(\hat{\alpha}_3 = 47.33 - 50.25\)- \(\hat{\alpha}_4 = 48.67 - 50.25\)For soil type (j) effects:- \(\hat{\beta}_1 = 53.75 - 50.25\)- \(\hat{\beta}_2 = 47 - 50.25\)- \(\hat{\beta}_3 = 50 - 50.25\)Calculate each of these estimated values.

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Key Concepts

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

Corrosion Protection

Corrosion protection is a crucial consideration in materials engineering, especially for metal components like pipes subjected to varying environmental conditions. Corrosion is the deterioration of materials through chemical interactions, typically with oxygen and moisture, leading to rust in metals. In our exercise, the effectiveness of four different coatings for preventing corrosion of metal pipes is under evaluation.

These coatings act as a barrier, preventing harmful elements that cause corrosion from reaching the metal surface. This process helps in extending the lifespan of the pipes significantly. By comparing the depth of corrosion for different coatings, we aim to identify the most effective option for protection. In summary, understanding corrosion protection is vital as it influences maintenance costs, safety, and overall durability of metal structures.

Experimental Design

Experimental design refers to the framework of organizing tests or experiments to ensure the results are valid and reliable. It's a systematic approach used to determine cause-and-effect relationships, and in our case, whether the type of coating or soil type substantially affects corrosion rates.

A good experimental design in this context includes:

• Selecting appropriate coatings and soil types for testing
• Randomly assigning the coatings to the pipes
• Standardizing the duration and conditions under which tests are conducted
• Repetition, to ensure reproducibility

These elements help minimize biases and errors. An essential aspect of this design is the use of the additive model, where we assume no interaction effect between coatings and soil types, allowing for the independent assessment of each factor.

Treatment Means

Treatment means represent the average outcomes for different treatment groups in an experiment. In our analysis, treatments refer to the different coatings applied to the pipes.

Calculating treatment means involves finding the average corrosion depth for each coating type. This step is vital as it allows us to see which coating performs best in preventing corrosion, by having the lowest mean corrosion depth.

• For instance, we computed means like:
• \( \mu_1 = 54.33 \) for Coating 1
• \( \mu_2 = 50.67 \) for Coating 2
• And so on for the rest

These differences in means reflect the coatings' effectiveness and set the foundation for further statistical analysis using ANOVA.

Sums of Squares

Sums of squares are a statistical tool used in ANOVA to quantify variation within an experiment. This concept helps in understanding how much of the total variability in data can be attributed to the differences in treatments and random errors.

There are different types of sums of squares in this context:

• **Total Sum of Squares (SST):** Reflects the overall variation in the data from the overall mean.
• **Treatment Sum of Squares (SSTR):** Measures the variation due to the coating treatments.
• **Soil Sum of Squares (SSBS):** Captures variation due to different soil types.
• **Error Sum of Squares (SSE):** Residual variation not explained by treatments or soil types.

By calculating these sums, we gain insight into where the most significant differences occur, enabling us to analyze the data's structure more effectively.

F-statistics

In ANOVA, the F-statistic is crucial for hypothesis testing, allowing us to assess whether differences among group means are statistically significant. It is essentially a ratio of variances that tells us if a factor contributes significantly to the data's variability.

Here's how it's calculated:

• **Mean Square (MS):** First, calculate MS for each source by dividing the sum of squares by associated degrees of freedom.
• **F-statistic:** Ratio of MS between group (treatment or soil type) to MS within group (error). For example, \( F_{Coating} = \frac{MS_{Coating}}{MS_{Error}} \)

An F-statistic larger than the critical value from F-distribution tables indicates significant differences among the treatment means at a given level of significance \( \alpha = 0.05 \). This helps in deciding whether to reject the null hypotheses for coating and soil types.