To fill in the missing values in the ANOVA table and compute the required P-value at a significance level of 0.05, we need to determine a few quantities based on the information provided.
Step 1: Calculate the sample sizes, means, and other necessary sums.
Groups:
- Group 1 (Menu-driven): 6 observations
- Group 2 (Command-driven): 6 observations
- Group 3 (Mixed): 6 observations
From the given data, we can find the sum of the times (add each group's values):
- Group 1: 12 + 16 + 10 + 12 + 10 + 16 = 76
- Group 2: 14 + 10 + 14 + 12 + 10 + 14 = 84
- Group 3: 10 + 8 + 10 + 10 + 8 + 4 = 60
Now we compute the total sum and the grand mean:
- Total Sum = 76 + 84 + 60 = 220
- Overall Count (N) = 18
- Grand Mean (G) = Total Sum / N = 220 / 18 ≈ 12.2222
Step 2: Calculate the sum of squares
- Treatment (Between Groups) Sum of Squares (SST): \[ SST = \sum \frac{n_i (\bar{x}_i - G)^2}{n} \] Where:
- \( \bar{x}_1 = 76/6 \approx 12.6667 \)
- \( \bar{x}_2 = 84/6 \approx 14 \)
- \( \bar{x}_3 = 60/6 \approx 10 \)
Calculating \( SST \): \[ SST = 6 \left( (12.6667 - 12.2222)^2 + (14 - 12.2222)^2 + (10 - 12.2222)^2 \right) \] \[ SST = 6 \left( (0.4445)^2 + (1.7778)^2 + (-2.2222)^2 \right) \] \[ SST = 6 \left( 0.1972 + 3.1605 + 4.9377 \right) \approx 6 \times 8.2954 \approx 49.7724 \]
Step 3: Calculate the error (within groups) sum of squares (SSE)
The formula for the Within Groups Sum of Squares (SSE): \[ SSE = \sum (x_{ij} - \bar{x}_i)^2 \]
Calculating within each group:
-
For Group 1: \[ (12 - 12.6667)^2 + (16 - 12.6667)^2 + (10 - 12.6667)^2 + (12 - 12.6667)^2 + (10 - 12.6667)^2 + (16 - 12.6667)^2 \] ≈ 2.7778 + 11.1111 + 7.1111 + 0.4444 + 7.1111 + 11.1111 = 39.6667
-
For Group 2: \[ (14 - 14)^2 + (10 - 14)^2 + (14 - 14)^2 + (12 - 14)^2 + (10 - 14)^2 + (14 - 14)^2 \] ≈ 0 + 16 + 0 + 4 + 16 + 0 = 36
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For Group 3: \[ (10 - 10)^2 + (8 - 10)^2 + (10 - 10)^2 + (10 - 10)^2 + (8 - 10)^2 + (4 - 10)^2 \] ≈ 0 + 4 + 0 + 0 + 4 + 36 = 44
Total SSE = 39.6667 + 36 + 44 = 119.6667
Step 4: Total Sum of Squares (SSTotal)
\[ SSTotal = SST + SSE \approx 49.7724 + 119.6667 = 169.4391 \]
Step 5: Fill in the ANOVA table
- Treatments Degrees of Freedom (df) = k - 1 = 3 - 1 = 2
- Error Degrees of Freedom (df) = N - k = 18 - 3 = 15
- Total Degrees of Freedom (df) = N - 1 = 18 - 1 = 17
Step 6: Calculate Mean Squares (MS)
- Treatments Mean Square (MST) = SST / df = 49.7724 / 2 ≈ 24.8862
- Error Mean Square (MSE) = SSE / df = 119.6667 / 15 ≈ 7.9786
Step 7: Calculate Fcalc
\[ F_{calc} = \frac{MST}{MSE} = \frac{24.8862}{7.9786} \approx 3.12 \]
Step 8: P-value determination
For F-distribution with (2, 15) degrees of freedom at a significance level of 0.05, we need to refer to F distribution tables or use statistical software to find P.
The critical value at 0.05 significance level (Ftab) is often around 4.0 for (2, 15) df. Since the calculated F is lower than the critical value, we fail to reject the null hypothesis.
Thus, the P-value would be greater than 0.05, making P not statistically significant.
Therefore, you can enter:
- \( P > 0.05 \)
Make sure you verify the F-value via a statistical calculator to properly get the exact P-value for your condition. It should closely align with general distributions!
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