Let's go through each blank (a-f) in the ANOVA table to determine the missing values.

Given Information:

1. Sum of Squares (SS) for Between Groups = 90
2. Degrees of Freedom (df) for Between Groups = 3
3. Sum of Squares (SS) for Within Groups = $d$
4. Degrees of Freedom (df) for Within Groups = 20
5. Total Sum of Squares (SS) = 210
6. Total Degrees of Freedom (df) = $f$

Calculations

Step 1: Fill in $d$ and $f$

1. Since Total SS = Between Groups SS + Within Groups SS: $210 = 90 + d$ $d = 210 - 90 = 120$

2. Since Total df = df (Between Groups) + df (Within Groups): $f = 3 + 20 = 23$

Step 2: Calculate Mean Squares (MS) for Between and Within Groups (a and e)

1. Mean Square for Between Groups (a): $\text{MS (Between Groups)} = \frac{\text{SS (Between Groups)}}{\text{df (Between Groups)}} = \frac{90}{3} = 30$ So, $a = 30$.

2. Mean Square for Within Groups (e): $\text{MS (Within Groups)} = \frac{\text{SS (Within Groups)}}{\text{df (Within Groups)}} = \frac{120}{20} = 6$ So, $e = 6$.

Step 3: Calculate F-ratio (b)

The F-ratio is calculated as: $\text{F-ratio} = \frac{\text{MS (Between Groups)}}{\text{MS (Within Groups)}} = \frac{30}{6} = 5$ So, $b = 5$.

Step 4: Determine $p$-Value (c)

To determine $c$ (the $p$-value), we need to consult an F-distribution table or use statistical software. However, with an F-ratio of 5, df1 = 3 (Between Groups), and df2 = 20 (Within Groups), we can approximate the $p$-value.

For an F-ratio of 5 with df1 = 3 and df2 = 20, the $p$-value is generally found to be around 0.01 or lower, indicating statistical significance. Thus, we can approximate: $c \approx 0.01$

Summary of Answers:

• a = 30
• b = 5
• c ≈ 0.01
• d = 120
• e = 6
• f = 23

Would you like further details or have questions about the calculations?

Related Questions:

1. What does the F-ratio represent in ANOVA analysis?
2. How do you interpret a significant $p$-value in ANOVA?
3. What are the assumptions of an ANOVA test?
4. How would the results change if the SS values were different?
5. Can ANOVA be used for comparing more than three groups?

Tip: In ANOVA, a low $p$-value indicates a statistically significant difference between group means.