Inferential Statistics: Single Factor ANOVA

Purpose: ANOVA is very similar to the t-Test and serves the function of determining statistical differences between group means.
Comparison with t-Test:
- t-Test: Examines whether there is a statistical difference between two means (Dependent Variable, DV) across two categories of an Independent Variable (IV) which is nominal and dichotomous.
- ANOVA: Used when comparing three or more group means (DV) across three or more categories of an IV that can be nominal, non-dichotomous, or ordinal, applying variance in the testing process.

Important Components of the ANOVA Test:
- Grand Mean vs. Group Means: The overall mean of all data points compared to individual group means.
- Within Group Variation vs. Between Group Variation:
- Within Group Variation: Variation within each individual group.
- Between Group Variation: Variation between the means of different groups.
- The F and the F Distribution: The ratio obtained in ANOVA to determine statistical significance.
- Degrees of Freedom: Number of independent pieces of information in analysis.
- Completing the Analysis: Involves using post hoc tests after ANOVA to further assess which means are significantly different.

Objective: To determine if there is a statistical difference in the mean number of MPH over the speed limit for three groups categorized by age: 24 and under, 25-34, and 35 and over.
Hypotheses:
- H1 (Alternative Hypothesis): There is a statistical difference in the overall mean scores across the three groups.
- H0 (Null Hypothesis): There is not a statistical difference in the overall mean scores across the three groups.
Data Summary:
- Groups:
- 24 and Under: 78 individuals
- 25 to 34: 75 individuals
- 35 and Over: 78 individuals
- Grand Mean: 11.61472
- Standard Error: 0.212923
- Sum Average Variance per group:
- 24 and Under: 12.48718 (Variance: 15.11022)
- 25 to 34: 10.91026 (Variance: 8.030803)
- 35 and Over: 11.44 (Variance: 7.114595)
- Median: 11
- Mode: 10
- Standard Deviation: 3.236148
- Sample Variance: 10.47265
- Kurtosis: 0.33254
- Skewness: 0.730164
- Range: 15 (Minimum: 6, Maximum: 21)
- Sum: 2683
- Count: 231

The Grand Mean is calculated via
$ext{Grand Mean} = rac{X<em>1 + X</em>2 + X_3}{3}$
- Where $X1$, $X2$, $X_3$ are the group means for the respective age groups involved.
Between Group Variation & Within Group Variation are key metrics calculated for F-ratio determination.

Meaning of F:
- The F statistic represents the ratio of the mean square values derived from within-group variance and between-group variance.
Interpreting F:
- If the null hypothesis (H0) is true, the ratio should hover close to 1. The higher the calculated F ratio, the stronger the evidence against H0 and the greater likelihood that the differences in group means are significant.

Degrees of Freedom:
- Between DF = Number of groups - 1
- Within DF = Sample Size - Number of groups
- The F-ratio is computed as
  $F = rac{ ext{Between MS}}{ ext{Within MS}}$
P-Value:
- Represents the alpha, typically set at 0.05 for a cutoff of 95% confidence.
- If p-value ≤ 0.05, significant differences exist; otherwise, they do not.
F Crit: The critical value retrieved from the F distribution table, necessary for comparing calculated F against the critical F value to determine significance.

While the ANOVA test can establish if an overall difference exists, it does not pinpoint where these differences lie.
Post Hoc Analysis (Tukey-Kramer): Used to identify specific group differences after finding a statistically significant result in ANOVA.
Questions Addressed by Post Hoc Analysis:
- Is there a difference between the groups 24 and Under and 25-34?
- Is there a difference between 25-34 and 35 and Older?
- Is there a difference between 24 and Under and 35 and Older?

Indicates that even though there’s an overall significant difference, the only significant difference in means occurs between the groups 24 and Under versus 35 and Over.