ANOVA

Problems Involving ANOVA

Problem Overview

• Context: A researcher wants to test if different teaching methods affect student test scores.

• Teaching Methods:

  • Traditional

  • Online

  • Hybrid

• Data Sources:

  • Traditional Scores: 85, 88, 90, 93, 87

  • Mean ($x_1$): 88.6

  • Sum of Scores ($ ext{xi}$): 443

  • Online Scores: 78, 82, 85, 79, 81

  • Mean ($x_2$): 81

  • Sum of Scores ($ ext{xi}$): 405

  • Hybrid Scores: 90, 92, 94, 89, 91

  • Mean ($x_3$): 91.2

  • Sum of Scores ($ ext{xi}$): 456

Step-by-Step ANOVA Procedure

Step 1: Identify the Type of Test

• Test Used: One-Way ANOVA

Step 2: State the Hypotheses

• Null Hypothesis ($H_0$): There is no difference in mean test scores across the teaching methods.

• Alternative Hypothesis ($H_1$): At least one teaching method has a significantly different mean score.

Step 3: Calculate ANOVA/F-ratio

F-ratio Formula

• $F = \frac{MSB}{MSW}$

Calculating Mean Square Values

1. Mean Square Between Groups (MSB):

   • Formula: $MSB = \frac{SSB}{k-1}$

   • Where:

     • SSB = Between-Group Sum of Squares

     • $k$ = number of groups

2. Mean Square Within Groups (MSW):

   • Formula: $MSW = \frac{SSW}{N-k}$

   • Where:

     • SSW = Within-Group Sum of Squares

     • $N$ = total number of observations

Between-Group Sum of Squares (SSB)

• Formula: $SSB = \sum(n<em>i (\bar{x}</em>i - \bar{x})^2$

• Where:

  • $n_i$: number of observations in each group

  • $\bar{x}_i$: mean of each group

  • $\bar{x}$: overall mean

• Calculation:

  • Overall Mean ($\bar{x}$): $\bar{x} = \frac{443 + 405 + 456}{15} <br>ightarrow \bar{x} = 88.8$

  • $SSB = 5(88.6 - 88.8)^2 + 5(81 - 88.8)^2 + 5(91.2 - 88.8)^2$

  • Calculation:

  • $SSB = 5(-0.2)^2 + 5(-7.8)^2 + 5(2.4)^2$

  • $= 5(0.04) + 5(60.84) + 5(5.76)$

  • $= 0.2 + 304.2 + 28.8 = 333.2$

• Final Calculation of SSB:

  • SSB value presented in transcript: $SSB = 280.9335$ (Note that there are corrections needed)

Within-Group Sum of Squares (SSW)

• Formula: $SSW = \sum{(x<em>{ij} - \bar{x}</em>i)^2}$

• Detailed calculations for each group:

  • Traditional Group:

  • For each score, calculate $(x<em>{ij} - \bar{x}</em>1)^2$

  • 85: $(85 - 88.6)^2 = 12.96$

  • 88: $(88 - 88.6)^2 = 0.36$

  • 90: $(90 - 88.6)^2 = 1.96$

  • 93: $(93 - 88.6)^2 = 19.36$

  • 87: $(87 - 88.6)^2 = 2.56$

  • Sum for Traditional: $(12.96 + 0.36 + 1.96 + 19.36 + 2.56) = 37.2$

  • Online Group:

  • 78: $(78 - 81)^2 = 9$

  • 82: $(82 - 81)^2 = 1$

  • 85: $(85 - 81)^2 = 16$

  • 79: $(79 - 81)^2 = 4$

  • 81: $(81 - 81)^2 = 0$

  • Sum for Online: $(9 + 1 + 16 + 4 + 0) = 30$

  • Hybrid Group:

  • 90: $(90 - 91.2)^2 = 1.44$

  • 92: $(92 - 91.2)^2 = 0.64$

  • 94: $(94 - 91.2)^2 = 7.84$

  • 89: $(89 - 91.2)^2 = 4.84$

  • 91: $(91 - 91.2)^2 = 0.04$

  • Sum for Hybrid: $(1.44 + 0.64 + 7.84 + 4.84 + 0.04) = 14.8$

• Total SSW:

  • $SSW = 37.2 + 30 + 14.8 = 82.0$

Mean Square Values Calculation

• Calculating MSB:

  • $MSB = \frac{SSB}{k-1} = \frac{280.9335}{2} = 140.46675$

• Calculating MSW:

  • $MSW = \frac{SSW}{N-k} = \frac{82}{12} = 6.8333$

Final Calculation of F-ratio

• $F = \frac{MSB}{MSW} = \frac{140.46675}{6.8333} = 20.5562$

Step 4: Determine the Degrees of Freedom

• Degrees of Freedom Between Groups ($df_{between}$):

  • Formula: $df_{between} = k-1 = 3-1 = 2$

• Degrees of Freedom Within Groups ($df_{within}$):

  • Formula: $df_{within} = N-k = 15-3 = 12$

Step 5: Determine the Critical Value of F

• Critical Value:

  • Given $F_{critical} = 3.89$

Step 6: Compare and Make a Decision

• Comparison:

  • If F{critical} < F{ratio} then reject $H_0$

  • Here, 3.89 < 20.5562

• Decision:

  • Reject $H_0$

Step 7: Interpret the Results

• Conclusion:

  • There is significant evidence to conclude that at least one teaching method has a different effect on student test scores.

Key Formulas and Concepts Summary

• F-ratio: $F = \frac{MSB}{MSW}$

• Degrees of Freedom:

  • Between Groups: $df_{between} = k-1$

  • Within Groups: $df_{within} = N-k$

• Sum of Squares:

  • Between Groups: $SSB = \sum(n<em>i (\bar{x}</em>i - \bar{x})^2$

  • Within Groups: $SSW = \sum{(x<em>{ij} - \bar{x}</em>i)^2}$

• Decision Criteria:

  • Reject the null hypothesis if F{critical} < F{ratio}

Notes

• Ensure that calculations regarding SSB and SSW are double-checked for accuracy in the context of their respective groups.

• When interpreting results, emphasize both statistical significance and potential educational implications of findings.