Determine the missing values in the following one-way ANOVA table involving 4 treatments (groups) with sample sizes 15, 11, 8, and 13.
Do not round answers, but if you must round, report answers accurate to 4 decimal places.

SS
df	MS	   F   
Between
(Treatment)	201.0513			
Within
(Error)			
Total	891.2873	


To complete the one-way ANOVA table, we need to compute the missing values using the given data. The calculations proceed as follows:

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### Step 1: Determine degrees of freedom (df)
- **Degrees of freedom for Treatment (Between)**:  
  \( df_{\text{Between}} = k - 1 \),  
  where \( k \) is the number of groups (treatments).  
  \( df_{\text{Between}} = 4 - 1 = 3 \).

- **Degrees of freedom for Error (Within)**:  
  \( df_{\text{Within}} = N - k \),  
  where \( N \) is the total sample size.  
  Total sample size \( N = 15 + 11 + 8 + 13 = 47 \).  
  \( df_{\text{Within}} = 47 - 4 = 43 \).

- **Degrees of freedom for Total**:  
  \( df_{\text{Total}} = N - 1 = 47 - 1 = 46 \).  
  This agrees with \( df_{\text{Total}} = df_{\text{Between}} + df_{\text{Within}} = 3 + 43 = 46 \).

---

### Step 2: Compute the mean squares (MS)
- **Mean square for Treatment (MS\(_{\text{Between}}\))**:  
  \( MS_{\text{Between}} = \frac{SS_{\text{Between}}}{df_{\text{Between}}} \).  
  \( MS_{\text{Between}} = \frac{201.0513}{3} = 67.0171 \).

- **Mean square for Error (MS\(_{\text{Within}}\))**:  
  We will calculate this after finding \( SS_{\text{Within}} \) in the next step.

---

### Step 3: Compute \( SS_{\text{Within}} \)
- Total sum of squares (SS\(_{\text{Total}}\)):  
  \( SS_{\text{Total}} = SS_{\text{Between}} + SS_{\text{Within}} \).  
  \( 891.2873 = 201.0513 + SS_{\text{Within}} \).  
  \( SS_{\text{Within}} = 891.2873 - 201.0513 = 690.2360 \).

---

### Step 4: Compute MS\(_{\text{Within}}\)
- \( MS_{\text{Within}} = \frac{SS_{\text{Within}}}{df_{\text{Within}}} \).  
  \( MS_{\text{Within}} = \frac{690.2360}{43} = 16.0476 \).

---

### Step 5: Compute F-statistic
- \( F = \frac{MS_{\text{Between}}}{MS_{\text{Within}}} \).  
  \( F = \frac{67.0171}{16.0476} = 4.1773 \).

---

### Final Completed Table:

| Source          | SS         | df  | MS         | F       |
|------------------|------------|------|------------|---------|
| Between (Treatment) | 201.0513  | 3    | 67.0171   | 4.1773  |
| Within (Error)   | 690.2360   | 43   | 16.0476   |         |
| Total            | 891.2873   | 46   |            |         |

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Let me know if you'd like a detailed explanation of any part of the calculation or more insights!

#### Expanding Questions:
1. What is the interpretation of the F-statistic in this context?
2. How does the number of groups affect the degrees of freedom in the ANOVA table?
3. What are the assumptions required for performing a one-way ANOVA?
4. How would the results change if the sample sizes across groups were more unequal?
5. Can we determine statistical significance from the F-value alone? If not, what additional information is needed?

#### Tip:
Always verify your total \( df \) and \( SS \) values, as they serve as checks for the accuracy of your calculations!

Determine the missing values in the following one-way ANOVA table involving 4 treatments (groups) with sample sizes 15, 11, 8, and 13. Do not round answers, but if you must round, report answers accurate to 4 decimal places.

SS
df	MS	   F   
Between
(Treatment)	201.0513
  
Within
(Error)
  
Total	891.2873

Learn how to complete a one-way ANOVA table by calculating missing values for four treatments with sample sizes 15, 11, 8, and 13. The solution covers the degrees of freedom, mean squares, and F-statistic. This explanation is suitable for students in Grades 11-12 studying statistics.

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