To complete the given ANOVA table and test for significance, we need to calculate the necessary values from the data provided. The regression equation is already provided as:

$\hat{y} = 1246.67 + 7.6x$

Here's how to proceed step-by-step:

1. Calculate Total Sum of Squares (SST): $SST = \sum (y_i - \bar{y})^2$ where $\bar{y}$ is the mean of $y$ values.

2. Calculate Regression Sum of Squares (SSR): $SSR = \sum (\hat{y}_i - \bar{y})^2$ where $\hat{y}_i = 1246.67 + 7.6x_i$.

3. Calculate Error Sum of Squares (SSE): $SSE = \sum (y_i - \hat{y}_i)^2$ Note: $SST = SSR + SSE$.

4. Calculate Degrees of Freedom:

   • Regression: $df_{\text{Regression}} = 1$ (only one predictor).
   • Error: $df_{\text{Error}} = n - 2$ ($n$ is the number of data points).
   • Total: $df_{\text{Total}} = n - 1$.

5. Calculate Mean Squares:

   • $MSR = SSR / df_{\text{Regression}}$
   • $MSE = SSE / df_{\text{Error}}$

6. F-Statistic: $F = \frac{MSR}{MSE}$

7. P-value: Use the F-distribution to find the p-value with the corresponding degrees of freedom.

Let me compute the required values for the table. import pandas as pd import scipy.stats as stats

Given data

x = np.array([400, 450, 550, 600, 700, 750]) # Production Volume y = np.array([4000, 5000, 5400, 5900, 6400, 7000]) # Total Cost

Regression parameters

b0 = 1246.67 # Intercept b1 = 7.6 # Slope

Predicted y values

y_pred = b0 + b1 * x

Mean of observed y values

y_mean = np.mean(y)

SST (Total Sum of Squares)

SST = np.sum((y - y_mean) ** 2)

SSR (Regression Sum of Squares)

SSR = np.sum((y_pred - y_mean) ** 2)

SSE (Error Sum of Squares)

SSE = np.sum((y - y_pred) ** 2)

Degrees of Freedom

n = len(y) # Number of observations df_regression = 1 # Regression degrees of freedom df_error = n - 2 # Error degrees of freedom df_total = n - 1 # Total degrees of freedom

Mean Squares

MSR = SSR / df_regression # Mean square for regression MSE = SSE / df_error # Mean square for error

F-statistic

F_stat = MSR / MSE

p-value

p_value = 1 - stats.f.cdf(F_stat, df_regression, df_error)

Compile results into a table

anova_table = pd.DataFrame({ "Source of Variation": ["Regression", "Error", "Total"], "Degrees of Freedom": [df_regression, df_error, df_total], "Sum of Squares": [SSR, SSE, SST], "Mean Square": [MSR, MSE, None], "F": [F_stat, None, None], "p-value": [p_value, None, None] })

import ace_tools as tools; tools.display_dataframe_to_user(nameThe ANOVA table for the regression analysis has been calculated and displayed for your reference. Let me know if you would like further explanations or assistance with interpreting the results.

Here are some follow-up questions for you:

1. What conclusions can you draw about the significance of the production volume's effect on total cost at $\alpha = 0.05$?
2. How is the F-statistic used to determine the significance of the regression model?
3. Would a lower p-value (e.g., <0.001) change your interpretation of the relationship between production volume and total cost?
4. Can you describe how the degrees of freedom are determined in the ANOVA table?
5. Would the result differ if the data set was larger? Why or why not?

Tip: Always confirm the p-value against the significance level ($\alpha$) to validate your hypothesis testing conclusion.