Understanding the ANOVA Two-Way Classification Table

The problem provides an incomplete ANOVA table for a two-way classification analysis, likely examining the effect of two factors (e.g., Variety A and Variety B of cloth) on some response variable, alongside the effect of error. We are asked to find the missing values represented by \(\alpha\) and \(\beta\).

The structure of an ANOVA table shows how the total variation in the data is partitioned among different sources of variation. The key columns are:

• Source of Variation: The factors or effects being tested (Variety A, Variety B, Error, Total).
• SS (Sum of Squares): A measure of the variability due to each source.
• df (degrees of freedom): Related to the number of independent pieces of information used to calculate the sum of squares.
• MS (Mean Square): Calculated as \(MS = SS / df\). It represents the variance estimate for each source.
• F-Ratio: Calculated as \(F-Ratio = MS_{Source} / MS_{Error}\). Used to test the significance of each source of variation.

Let's look at the given table:

Calculating the Missing Values

Step 1: Find the value of \(\beta\) (df for Error)

The total degrees of freedom (\(df_{Total}\)) in an ANOVA table is the sum of the degrees of freedom for all individual sources of variation.

\(df_{Total} = df_{Variety\ A} + df_{Variety\ B} + df_{Error}\)

From the table, we have:

• \(df_{Total} = 11\)
• \(df_{Variety\ A} = 2\)
• \(df_{Variety\ B} = 3\)
• \(df_{Error} = \beta\)

Plugging these values into the formula:

\(11 = 2 + 3 + \beta\)

\(11 = 5 + \beta\)

Solving for \(\beta\):

\(\beta = 11 - 5\)

\(\beta = 6\)

We can also verify this using the Error row information. \(MS_{Error} = SS_{Error} / df_{Error}\). Given \(SS_{Error} = 20\) and \(MS_{Error} = 3.33\):

\(3.33 = 20 / \beta\)

\(\beta = 20 / 3.33\)

\(\beta \approx 6.006\)

This calculation confirms that \(\beta = 6\) is the correct degree of freedom for the error term, likely rounded in the MS value.

Step 2: Find the value of \(\alpha\) (SS for Variety B)

Similar to degrees of freedom, the total Sum of Squares (\(SS_{Total}\)) is the sum of the Sums of Squares for all sources of variation (assuming a fixed-effects model without interactions or if interactions are combined into error, based on the structure). In this simple two-way structure shown, it's typically the sum of main effects and error.

\(SS_{Total} = SS_{Variety\ A} + SS_{Variety\ B} + SS_{Error}\)

From the table, we have:

• \(SS_{Total} = 640\)
• \(SS_{Variety\ A} = 280\)
• \(SS_{Variety\ B} = \alpha\)
• \(SS_{Error} = 20\)

Plugging these values into the formula:

\(640 = 280 + \alpha + 20\)

\(640 = 300 + \alpha\)

Solving for \(\alpha\):

\(\alpha = 640 - 300\)

\(\alpha = 340\)

Let's quickly check consistency with other values in the table, although the primary way to find \(\alpha\) and \(\beta\) from the given options and table structure is usually via the sum properties of SS and df. Using \(MS_{Variety\ B} = SS_{Variety\ B} / df_{Variety\ B}\): \(334.03 = \alpha / 3\) \(\alpha = 334.03 \times 3 \approx 1002.09\). This value is different from 340. Using \(MS_{Error} = 3.33\) and \(MS_{Variety\ A} = 140\), the F-Ratio for Variety A is \(140 / 3.33 \approx 42.04\), which matches the table. This confirms MS_Error is correct and derived from \(SS_{Error}\) and \(df_{Error}\) where \(df_{Error}=6\). The total SS and total df calculations lead to \(\alpha = 340\) and \(\beta = 6\). The value of MS for Variety B (334.03) seems inconsistent with the calculated \(\alpha=340\) and \(df_{Variety\ B}=3\) (\(340/3 \approx 113.33\)). However, given the options, the values derived from the sum properties of SS and df (\(\alpha=340, \beta=6\)) are available as an option. We will proceed with these values as they are consistent with the total SS and total df provided.

The Values of \(\alpha\) and \(\beta\)

Based on our calculations using the relationships within the ANOVA table:

• \(\alpha = 340\)
• \(\beta = 6\)

So, the values of \((\alpha, \beta)\) are \((340, 6)\).

Revision Table: Key ANOVA Terms

Additional Information: ANOVA Concepts

ANOVA (Analysis of Variance) is a statistical technique used to compare the means of three or more groups simultaneously. It does this by partitioning the total variability in a dataset into different components attributed to different sources of variation.

• One-Way ANOVA: Used when comparing means across different levels of a single factor.
• Two-Way ANOVA: Used when comparing means across different levels of two factors. It can also test for an interaction effect between the two factors. The table structure in this problem is a simplified two-way ANOVA, possibly without an explicit interaction term shown separately.
• Hypothesis Testing in ANOVA: For each source of variation (except Error), an F-test is performed.
  • Null Hypothesis (\(H_0\)): There is no significant difference in means across the levels of this factor (or no interaction effect).
  • Alternative Hypothesis (\(H_1\)): There is a significant difference in means (or there is an interaction effect).
  The calculated F-Ratio is compared to a critical F-value from the F-distribution table or a p-value is obtained. If the calculated F-Ratio is large (or p-value is small), we reject \(H_0\).
• Error Term: The variability within each group or due to random chance is captured by the error term. The Mean Square Error (\(MS_{Error}\)) is often considered an estimate of the population variance \(\sigma^2\).

Understanding the relationships between SS, df, MS, and F-Ratio is crucial for interpreting ANOVA tables and performing statistical tests on the effects of different factors.