MAS183 — Pre-Semester Study Guide

Foundations: Data & Measurement

MASREF 1

1A. Types of Data
1B. Measurement Level

Describing Data

MASREF 2

2A. Measures of Location
2B. Measures of Spread
2C. Measures of Shape

Sampling & Bias

MASREF 3

Sampling Methods

Types of Bias

• Self-selection bias: over-represents people with strong opinions
• Convenience bias: harder-to-reach parts of population are excluded
• Non-response bias: those who opt out differ systematically from participants
• Investigator intervention: humans are poor at selecting a truly "average" group

Key distinction: Bias is a property of the sampling process; representation describes how well the resulting sample reflects the population.

Probability

MASREF 4

Core Vocabulary

• Sample space (S): all possible outcomes
• Event (A): a particular outcome or set of outcomes
• Complementary event (A'): A does not occur; P(A') = 1 − P(A)
• Values for probabilities: always between 0 (impossible) and 1 (certain)

Probability Rules

• Proportion of outcomes: P(A) = number of favourable outcomes / total outcomes
• Addition rule:
  • General: P(A or B) = P(A) + P(B) − P(A and B)
  • Mutually exclusive events: P(A or B) = P(A) + P(B)
• Multiplication rule:
  • General: P(A and B) = P(A) × P(B | A)
  • Independent events: P(A and B) = P(A) × P(B)

Key Event Types

• Mutually exclusive: cannot occur together; P(A and B) = 0
• Independent events: occurrence of one does not affect the other
• Joint event (A and B): both events occur together
• Conditional probability P(A | B): chance of A occurring given B has occurred; note the denominator changes

MASREF 5 — Random Variables & Probability Distributions

Random Variables

• Random variable (RV): a variable whose value is determined by a random process
• Discrete RV: takes countable values (e.g., number of successes)
• Continuous RV: can take any value in a range; probability of any exact value = 0

Defining a Probability Distribution

1. Enumerate all possible values and their probabilities
2. Tabulate values and probabilities
3. Give the formula for P(X = x)
4. State name and parameters in standard notation (if a named distribution)

Types of Probability

• Individual probability: P(X = x) — chance of one specific value
• Cumulative probability: P(X ≤ x) — accumulates all probabilities up to x

MASREF 6 — Named Distributions

Binomial Distribution

Applies when all five conditions are met:

1. Fixed number of trials, n
2. Each trial has exactly two outcomes: success or failure
3. Probability of success, p, is constant across trials
4. Trials are independent
5. Variable of interest, X, counts the number of successes

Normal Distribution

Key features:

• Bell-shaped and symmetric around the mean
• Mean = median
• Asymptotic tails (approach zero but never reach it)
• Described by two parameters: mean (μ) and standard deviation (σ)

Standard Normal Distribution

• Special case: mean = 0, standard deviation = 1
• Z-score: measures how many standard deviations a value lies from the mean
  • z = (X − μ) / σ
• Used to find probabilities for any Normal distribution

MASREF 7 — Sampling Distributions & Inference

From Sample to Population

• Population parameter: a characteristic of the whole population (usually unknown)
• Point estimate: a single sample value used to estimate a parameter
• Population proportion: ratio of members with a characteristic to total population

Central Limit Theorem (CLT)

For samples of size n drawn from any population distribution, the sampling distribution of the sample mean approaches a Normal distribution as n increases. This is what makes inference possible.

Confidence Intervals

• Confidence interval: a range of values likely to contain the true population parameter
• Confidence level: the estimated probability that the interval captures the parameter
• Wider intervals = more confidence; narrower intervals = more precision

Hypothesis Testing

• P-value: the probability of observing a result at least as extreme as the one obtained, assuming the null hypothesis is true
• Small p-value → evidence against the null hypothesis

MASREF 8 — Correlation & Regression

Describing Bivariate Relationships (from scatterplots)

• Direction: positive or negative trend
• Shape: linear or curved
• Strength: how tightly points cluster around the trend line

Correlation

• Measures the strength and direction of a linear relationship between two numerical variables
• Ranges from −1 (perfect negative) to +1 (perfect positive)

Regression

• Explanatory (predictor) variable: x — used to predict y
• Response variable: y — the outcome being predicted
• Least-squares estimation: fits a line by minimising the sum of squared errors
• Slope: change in y for each 1-unit increase in x
• Y-intercept: value of y when x = 0

Coefficient of Determination (R²)

• The percentage of variation in y explained by variation in x
• R² close to 100% → x is a strong predictor of y
• R² close to 0% → x tells us very little about y