This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Hey faith, good to see you again. Here are the steps to solve the problem: Part i: State the null and alternate hypotheses for the question above. Null Hypothesis (H_0): There is no significant difference in the mean scores among students in Agric Education, Business Education, and Industrial Technical Education. H_0: _1 = _2 = _3 Alternate Hypothesis (H_1): At least one group mean score is significantly different from the others. H_1: Not all means are equal Part ii: State the level of significance. The level of significance is not explicitly stated in the question. We will assume a common level of significance. Level of Significance (): 0.05 Part iii: Calculate the F-value. We will perform a one-way ANOVA to calculate the F-value. Step 1: Calculate the sum, mean, and sum of squares for each group. Agric Education (Group 1): Scores (X1) = 7, 6, 6, 4, 8, 3, 6, 5, 7, 4 Number of observations (n_1) = 10 Sum (T_1) = 7+6+6+4+8+3+6+5+7+4 = 56 Mean (X_1) = (56)/(10) = 5.6 Sum of squares ( X_1^2) = 7^2+6^2+6^2+4^2+8^2+3^2+6^2+5^2+7^2+4^2 = 336 Business Education (Group 2): Scores (X2) = 5, 4, 5, 6, 3, 5, 4, 7, 3, 4 Number of observations (n_2) = 10 Sum (T_2) = 5+4+5+6+3+5+4+7+3+4 = 46 Mean (X_2) = (46)/(10) = 4.6 Sum of squares ( X_2^2) = 5^2+4^2+5^2+6^2+3^2+5^2+4^2+7^2+3^2+4^2 = 230 Industrial Technical Education (Group 3): Scores (X3) = 8, 5, 6, 9, 7, 8, 9, 5, 6, 8 Number of observations (n_3) = 10 Sum (T_3) = 8+5+6+9+7+8+9+5+6+8 = 71 Mean (X_3) = (71)/(10) = 7.1 Sum of squares ( X_3^2) = 8^2+5^2+6^2+9^2+7^2+8^2+9^2+5^2+6^2+8^2 = 525 Step 2: Calculate the total number of observations, grand sum, and total sum of squares. Total number of observations (N) = n_1 + n_2 + n_3 = 10 + 10 + 10 = 30 Grand Sum (G) = T_1 + T_2 + T_3 = 56 + 46 + 71 = 173 Total Sum of Squares of all scores ( X^2) = X_1^2 + X_2^2 + X_3^2 = 336 + 230 + 525 = 1091 Step 3: Calculate the Correction Factor (CF). CF = (G^2)/(N) = (173^2)/(30) = (29929)/(30) = 997.6333 Step 4: Calculate the Total Sum of Squares (SST). SST = X^2 - CF = 1091 - 997.6333 = 93.3667 Step 5: Calculate the Sum of Squares Between Groups (SSB). SSB = ( (T_1^2)/(n_1) + (T_2^2)/(n_2) + (T_3^2)/(n_3) ) - CF SSB = ( (56^2)/(10) + (46^2)/(10) + (71^2)/(10) ) - 997.6333 SSB = ( (3136)/(10) + (2116)/(10) + (5041)/(10) ) - 997.6333 SSB = (313.6 + 211.6 + 504.1) - 997.6333 SSB = 1029.3 - 997.6333 = 31.6667 Step 6: Calculate the Sum of Squares Within Groups (SSW). SSW = SST - SSB = 93.3667 - 31.6667 = 61.7000 Step 7: Calculate the Degrees of Freedom. Degrees of Freedom Between Groups (df_1) = k - 1 = 3 - 1 = 2 (where k is the number of groups) Degrees of Freedom Within Groups (df_2) = N - k = 30 - 3 = 27 Step 8: Calculate