Transcription for Educational Website:---Title: Understanding Power Curves and Experimental Data Analysis### Power Curves for the ANOVA Test ($v_1 = 4$)- **Graph Explanation:** - The graph displays power curves for the ANOVA test with $v_1 = 4$. - The x-axis represents the non-centrality parameter $\lambda$. - The y-axis represents the power (1 - $\beta$), where $\beta$ is the Type II error probability. - Curves are plotted for different values of significance level $\alpha$ (e.g., 0.05, 0.01). - The graph helps visualize the relationship between power, sample size, effect size, and significance level.### Experimental Data Analysis#### (a) Corrosion Experiment- Twelve pieces of pipe are selected, each coated with one of four coatings, and buried in one of three types of soil for a fixed time.- Objectives: - Find $b$ for various conditions where $c_1 = 4$, $c_2 = 0.5$, $\alpha = 2$, and $\alpha = 4$. - Repeat calculations for $c_1 = 6$, $c_2 = 0.75$, $\alpha = 3$, and $\alpha = 4$. - Use $\alpha = 0.01$. Answers should be given to two decimal places.#### (b) Fabric Experiment- This experiment aims to compare three different brands of pens and four different washing treatments.- The response variable is a quantitative measure of their ability to remove marks on a specific type of fabric.##### Data Table:| Washing Treatment | Brand of Pen | Total Average ||-------------------|--------------|---------------|| | 1 | 2 | 3 | || Total Average | 0.083 | 1.337 | 0.300 | || Average | 0.97 | 0.57 | 0.66 | |- **Interpretation:** - Lower values in the table indicate more marks removed. - This analysis helps determine the effectiveness of each combination of pen and washing treatment.### Symmetrical Calculations- Calculate $b\ **Power Curves for ANOVA Test (ν₁ = 3)**The image contains two graphs titled "Power Curves for the ANOVA Test (ν₁ = 3)" which show the relationship between power (1 - β) and the non-centrality parameter δ for different significance levels.**Graph Specifications:**1. **Axes:** - The horizontal axis represents the non-centrality parameter, denoted as \(\delta$. - The vertical axis represents the power of the test, denoted as $1 - \beta$.2. **Curves:** - Each graph is composed of several curves. Each curve corresponds to a different degree of freedom ($\nu_2$) and ranges from 1 to $5$. - Two main levels of significance ($\alpha$) are represented: $\alpha = 0.1$ and $\alpha = 0.05$.3. **Graphs:** - **Left Graph:** - Displays power curves for $\alpha = 0.1$. - The curves depict the relationship between power and the non-centrality parameter for different values of $\nu_2$. - **Right Graph:** - Displays power curves for $\alpha = 0.05$. - Similar layout to the left graph, focusing on power across the non-centrality parameter for different $\nu_2$ values.**Text Explanation:**The power curves in the figures can be used to determine the probability of correctly rejecting a false null hypothesis (Power = $1 - \text{Type II Error}$) for the F-test in two-factor ANOVA. For fixed values of $\alpha$, $\nu_2$, etc., the non-centrality parameter δ can be computed. The power is read on the vertical axis from the curve labeled by $z = (1/\Delta)\left(\Sigma n_i z_i - 1\right)$.These graphs thus provide insights into how different significance levels and degrees of freedom in ANOVA tests can impact the power to detect true effects.

Note: Since you have posted multiple questions, we will provide the solution only to the first…

Answered: (a) For a corrosion experiment, α₁ = 4,…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Related questions

Question

$Transcription for Educational Website: --- Title: Understanding Power Curves and Experimental Data Analysis ### Power Curves for the ANOVA Test ($v_1 = 4$) - **Graph Explanation:** - The graph displays power curves for the ANOVA test with $v_1 = 4$. - The x-axis represents the non-centrality parameter $\lambda$. - The y-axis represents the power (1 - $\beta$), where $\beta$ is the Type II error probability. - Curves are plotted for different values of significance level $\alpha$ (e.g., 0.05, 0.01). - The graph helps visualize the relationship between power, sample size, effect size, and significance level. ### Experimental Data Analysis #### (a) Corrosion Experiment - Twelve pieces of pipe are selected, each coated with one of four coatings, and buried in one of three types of soil for a fixed time. - Objectives: - Find $b$ for various conditions where $c_1 = 4$, $c_2 = 0.5$, $\alpha = 2$, and $\alpha = 4$. - Repeat calculations for $c_1 = 6$, $c_2 = 0.75$, $\alpha = 3$, and $\alpha = 4$. - Use $\alpha = 0.01$. Answers should be given to two decimal places. #### (b) Fabric Experiment - This experiment aims to compare three different brands of pens and four different washing treatments. - The response variable is a quantitative measure of their ability to remove marks on a specific type of fabric. ##### Data Table: | Washing Treatment | Brand of Pen | Total Average | |-------------------|--------------|---------------| | | 1 | 2 | 3 | | | Total Average | 0.083 | 1.337 | 0.300 | | | Average | 0.97 | 0.57 | 0.66 | | - **Interpretation:** - Lower values in the table indicate more marks removed. - This analysis helps determine the effectiveness of each combination of pen and washing treatment. ### Symmetrical Calculations - Calculate \(b\$

Transcribed Image Text:Transcription for Educational Website: --- Title: Understanding Power Curves and Experimental Data Analysis ### Power Curves for the ANOVA Test ($v_1 = 4$) - **Graph Explanation:** - The graph displays power curves for the ANOVA test with $v_1 = 4$. - The x-axis represents the non-centrality parameter $\lambda$. - The y-axis represents the power (1 - $\beta$), where $\beta$ is the Type II error probability. - Curves are plotted for different values of significance level $\alpha$ (e.g., 0.05, 0.01). - The graph helps visualize the relationship between power, sample size, effect size, and significance level. ### Experimental Data Analysis #### (a) Corrosion Experiment - Twelve pieces of pipe are selected, each coated with one of four coatings, and buried in one of three types of soil for a fixed time. - Objectives: - Find $b$ for various conditions where $c_1 = 4$, $c_2 = 0.5$, $\alpha = 2$, and $\alpha = 4$. - Repeat calculations for $c_1 = 6$, $c_2 = 0.75$, $\alpha = 3$, and $\alpha = 4$. - Use $\alpha = 0.01$. Answers should be given to two decimal places. #### (b) Fabric Experiment - This experiment aims to compare three different brands of pens and four different washing treatments. - The response variable is a quantitative measure of their ability to remove marks on a specific type of fabric. ##### Data Table: | Washing Treatment | Brand of Pen | Total Average | |-------------------|--------------|---------------| | | 1 | 2 | 3 | | | Total Average | 0.083 | 1.337 | 0.300 | | | Average | 0.97 | 0.57 | 0.66 | | - **Interpretation:** - Lower values in the table indicate more marks removed. - This analysis helps determine the effectiveness of each combination of pen and washing treatment. ### Symmetrical Calculations - Calculate \(b\

**Power Curves for ANOVA Test (ν₁ = 3)**

The image contains two graphs titled "Power Curves for the ANOVA Test (ν₁ = 3)" which show the relationship between power (1 - β) and the non-centrality parameter δ for different significance levels.

**Graph Specifications:**

1. **Axes:**
- The horizontal axis represents the non-centrality parameter, denoted as $\delta$.
- The vertical axis represents the power of the test, denoted as $1 - \beta$.

2. **Curves:**
- Each graph is composed of several curves. Each curve corresponds to a different degree of freedom ($\nu_2$) and ranges from 1 to $5$.
- Two main levels of significance ($\alpha$) are represented: $\alpha = 0.1$ and $\alpha = 0.05$.

3. **Graphs:**
- **Left Graph:**
- Displays power curves for $\alpha = 0.1$.
- The curves depict the relationship between power and the non-centrality parameter for different values of $\nu_2$.
- **Right Graph:**
- Displays power curves for $\alpha = 0.05$.
- Similar layout to the left graph, focusing on power across the non-centrality parameter for different $\nu_2$ values.

**Text Explanation:**

The power curves in the figures can be used to determine the probability of correctly rejecting a false null hypothesis (Power = $1 - \text{Type II Error}$) for the F-test in two-factor ANOVA. For fixed values of $\alpha$, $\nu_2$, etc., the non-centrality parameter δ can be computed. The power is read on the vertical axis from the curve labeled by $z = (1/\Delta)\left(\Sigma n_i z_i - 1\right)$.

These graphs thus provide insights into how different significance levels and degrees of freedom in ANOVA tests can impact the power to detect true effects.

Expert Solution

Step by step

Solved in 4 steps with 8 images

SEE SOLUTION Check out a sample Q&A here

(a) For a corrosion experiment, α₁ = 4, α₂ = 0, α3 = α4 = -2, a3 a4 a1 first = second ß =