Key Concepts in Number Theory and Statistics
Number Theory and Modular Arithmetic
1. Define GCD (1 Mark)
GCD (Greatest Common Divisor) is the largest positive integer that divides two or more integers exactly.
2. Prime Factorization of 1330 (1 Mark)
1330 = 2 × 5 × 7 × 19
3. Define Residue (1 Mark)
A residue modulo m is any integer equivalent to another integer modulo m.
4. State the Chinese Remainder Theorem (1 Mark)
If the moduli are pairwise coprime, a system of congruences has a unique solution modulo the product of the moduli.
5. Define Fermat
Read MoreStatistical Analysis and APA Reporting Templates
Multiple Linear Regression Analysis
Purpose: Predict a continuous dependent variable using two or more predictors.
Assumptions: Linearity, independence, homoscedasticity, normality of residuals, and no multicollinearity.
How to Read:
- Model Summary: R²
- ANOVA: F and p
- Coefficients: B, β, t, and p
Interpretation Formula: F(df1, df2) = value, p = value, R² = value.
Effect Size: R² = .02 (small), .13 (medium), .26 (large).
APA Template for Reporting Model and Predictors:
ANOVA Table: Regression: F(2, 47) =
Read MoreEssential Probability and Statistics Principles
Counting Principles and Techniques
Fundamental Counting Principle
Fundamental Counting Principle: If a task or process is made up of stages with separate (distinct) choices at each stage, the total number of choices is m × n × p × …, where m is the number of choices at the first stage, n is the number of choices at the second stage, and so forth.
Tree Diagrams vs. Fundamental Counting Principle
Tree Diagrams: A tree diagram is a useful tool for organized counting that shows all possible outcomes
Read MoreStatistical Analysis Methods and Data Interpretation
Box Plot Analysis for Distribution Comparison
Box plots are used for comparing distributions between groups.
Key Components:
- Median: The line inside the box representing the typical value.
- Q1/Q3: The bottom and top of the box, respectively.
- IQR (Interquartile Range): Calculated as Q3 − Q1, representing the middle 50% of the data.
- Narrow IQR: Indicates low variability and good reproducibility.
- Wide IQR: Indicates high variability.
- Whiskers: Represent the overall non-outlier range.
- Outliers: Points outside
Research Methodology: Sampling, Testing, and Reporting
Part 1: Sampling Design and Sampling Procedure
In research, it is usually impossible, too expensive, or too time-consuming to collect data from every single individual in a population (referred to as a census). Instead, researchers select a smaller, representative subset of that population, known as a sample.
- Population (N): The total collection of all elements that share a common set of characteristics (e.g., all BBA students in a university).
- Sample (n): The actual group chosen from the population
One-Way ANOVA: Concepts, Assumptions, and Applications
Understanding One-Way Analysis of Variance (ANOVA)
One-Way Analysis of Variance (ANOVA) is a statistical technique used to compare the means of three or more independent groups based on a single independent variable (factor).
It is utilized to determine whether there is a significant difference between group averages. In simple terms, One-Way ANOVA helps identify whether different groups behave differently regarding a measurable outcome.
Examples of One-Way ANOVA
- Comparing marks of students from three