Calculate the sum, mean, and sum of squares for each group.

• Agric. Education ($G_1$): Scores (8, 5, 5, 9, 7, 8, 9, 5, 6, 8) $n_1 = 10$ $\sum x_1 = 70$ $\bar{x}_1 = \frac{70}{10} = 7$ $\sum x_1^2 = 514$
• Business Education ($G_2$): Scores (7, 6, 6, 4, 8, 3, 6, 5, 7, 4) $n_2 = 10$ $\sum x_2 = 56$ $\bar{x}_2 = \frac{56}{10} = 5.6$ $\sum x_2^2 = 336$
• Industrial Technical Education ($G_3$): Scores (5, 4, 5, 6, 3, 5, 4, 7, 3, 4) $n_3 = 10$ $\sum x_3 = 46$ $\bar{x}_3 = \frac{46}{10} = 4.6$ $\sum x_3^2 = 230$

Overall: $N = n_1 + n_2 + n_3 = 10 + 10 + 10 = 30$
Grand Sum ($\sum x$) = $70 + 56 + 46 = 172$
Grand Sum of Squares ($\sum x^2$) = $514 + 336 + 230 = 1080$

• Correction Factor: $$CF = \frac{(\sum x)^2}{N} = \frac{(172)^2}{30} = \frac{29584}{30} = 986.1333$$
• Total Sum of Squares: $$SST = \sum x^2 - CF = 1080 - 986.1333 = 93.8667$$
• Sum of Squares Between Groups: $$SSB = \left(\frac{(\sum x_1)^2}{n_1} + \frac{(\sum x_2)^2}{n_2} + \frac{(\sum x_3)^2}{n_3}\right) - CF$$ $$SSB = \left(\frac{70^2}{10} + \frac{56^2}{10} + \frac{46^2}{10}\right) - 986.1333$$ $$SSB = (490 + 313.6 + 211.6) - 986.1333 = 1015.2 - 986.1333 = 29.0667$$
• Sum of Squares Within Groups: $$SSW = SST - SSB = 93.8667 - 29.0667 = 64.8$$

• Degrees of freedom between groups ($df_B$) = $k - 1 = 3 - 1 = 2$
• Degrees of freedom within groups ($df_W$) = $N - k = 30 - 3 = 27$

• Mean Square Between Groups: $$MSB = \frac{SSB}{df_B} = \frac{29.0667}{2} = 14.53335$$
• Mean Square Within Groups: $$MSW = \frac{SSW}{df_W} = \frac{64.8}{27} = 2.4$$

The calculated F-value is $\boxed{6.06}$.

III. What is the F-table value?

Using $\alpha = 0.05$, $df_1 = df_B = 2$, and $df_2 = df_W = 27$:
The F-table value is $\boxed{3.35}$.

IV. What is your decision based on the results?

Since the calculated F-value (6.06) is greater than the critical F-table value (3.35), we reject the null hypothesis.

V. What is your conclusion?

There is sufficient evidence to conclude that there is a significant difference in the mean quiz scores among the students in Agric. Education, Business Education, and Industrial Technical Education departments at the 0.05 level of significance.