University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

274

Pages:

4
Author:

Niklas

Table of Data: Interval Frequency (F) Cumulative Frequency (C) Class Value (CV) (x - x̄) (x - x̄)^2 f(x - x̄)^2 cv^2 f * cv^2 (0-5] 10 10 2.5 -6.8 46.24 462.4 6.25 62.5 (5-15] 10 20 10.5 -1.2 1.44 14.4 110.25 1102.5 (15-25] 5 25 20.5 11.2 125.44 627.2 420.25 2101.25 Експортувати в Таблиці Calculations: N = 25 x̄ = (2.5 * 10 + 10.5 * 10 + 20.5 * 5) / 25 = 9.3 σ² = 1404 / 25 = 44.16 Formulas and Explanations: Conditional mean: ȳj = (1/Nj) * Σ(Yi,j) General mean: ȳ = (1/N) * Σ(Nj * ȳj) Marginal variance: σ² = (1/N) * Σ(Σ(yi,j - ȳ)²) Conditional variance: σ²j = (1/Nj) * Σ(yi,j - ȳj)²) Relationship between general variance and conditional variance: σ² = (1/N) * Σ(Nj * σ²j) + (1/N) * Σ(Nj * (ȳj - ȳ)²) Example 1: Groups: 2 groups (male, female) Variables: 1 quantitative variable (height) Sample sizes: N1 = 20, N2 = 30 Variance within groups: Vw = 45.7 Variance between groups: Vb = 66.69 Total variance: Vt = Vw + Vb = 112.39 Explanation: The higher expected variance between groups is due to the fact that males are generally taller than females. This conclusion can be inferred from the data. Example 2: Variance: σ² = 72.4 Variance within groups: Vw = 71.69 Variance between groups: Vb = 0.71 Explanation: In this case, the contribution of Vb to the marginal variance is negligible due to the similar group means (ȳ1 = 79.73, ȳ2 = 78.03). Variance Decomposition: It is a tool to study the extent to which groups diverge in terms of the mean with respect to a quantitative variable. η² = Vb / Vt η² ranges from 0 to 1 The closer η² is to 1, the more different the groups Table of Data: Case A B C 0 1 1 2 1 2 3 2 2 3 3 4 3 3 4 8 Calculations: Case A: Mean (x̄): (1 + 2 + 3 + 3) / 4 = 2.25 Variance (σ²): Σ(x - x̄)² / N = 5 / 4 = 1.25 Case B: Mean (x̄): (1 + 1 + 3 + 3) / 4 = 2 Variance (σ²): Σ(f * (x - x̄)²) / N = 4 / 4 = 1 Case C: Mean (x̄): (2 + 2 + 4 + 8) / 4 = 4 Variance (σ²): Σ(f * (x - x̄)²) / N = 24 / 4 = 6 Table of Data: xi f (xi - x̄) (xi - x̄)² f(xi - x̄)² 0 1 -2.5 6.25 6.25 1 3 -1.5 2.25 6.75 2 3 -0.5 0.25 0.75 3 3 0.5 0.25 0.75 4 1 1.5 2.25 2.25 8 1 5.5 30.25 30.25 Calculations: Mean (x̄): (0 + 1 * 3 + 2 * 3 + 3 * 3 + 4 * 1 + 8 * 1) / 12 = 2.5 Variance within groups (Vw): (1/12) * (1.25 + 3 + 2) = 2.75 Variance between groups (Vb): (1/12) * ((1.5 - 2.5)² + (2 - 2.5)² + (4 - 2.5)²) = 1.16 η²: Vb / (Vb + Vw) = 1.16 / (1.16 + 2.75) = 0.3 (not very strong)