Distributions and Hypothesis Testing

Duration: 15 min

Visual: Common Distributions

Normal Distribution        Binomial Distribution
        │                         │
        │     ╱╲                  │  ╱╲  ╱╲
        │    ╱  ╲                 │ ╱  ╲╱  ╲
        │   ╱    ╲                │╱        ╲
    ────┼──────────────       ────┼──────────────
        μ                         p
        
68% within 1σ              n trials, p probability
95% within 2σ              each trial independent

Key Concepts Table

Common Probability Distributions

Normal Distribution

• Bell-shaped, symmetric curve
• Defined by mean (μ) and standard deviation (σ)
• 68% of data within 1σ, 95% within 2σ, 99.7% within 3σ
• Used in many natural phenomena

Binomial Distribution

• Models number of successes in n independent trials
• Each trial has probability p of success
• Example: coin flips, pass/fail outcomes

Poisson Distribution

• Models count of events in fixed time/space
• Defined by parameter λ (average rate)
• Example: number of emails per hour, customer arrivals

Exponential Distribution

• Models time between events
• Related to Poisson distribution
• Example: time until next customer arrives

Hypothesis Testing

Null and Alternative Hypotheses

• H₀ (Null): No effect or difference exists
• H₁ (Alternative): Effect or difference exists

Type I and Type II Errors

• Type I Error: Reject H₀ when it's true (false positive)
• Type II Error: Fail to reject H₀ when it's false (false negative)

Significance Level (α)

• Probability of Type I error
• Common values: 0.05, 0.01
• If p-value < α, reject H₀

P-value

• Probability of observing data if H₀ is true
• Lower p-value = stronger evidence against H₀
• p-value < 0.05 typically considered significant

Common Statistical Tests

T-test

• Compares means of two groups
• Assumes normal distribution
• Types: independent, paired, one-sample

Chi-Square Test

• Tests independence between categorical variables
• Compares observed vs expected frequencies

ANOVA (Analysis of Variance)

• Compares means across 3+ groups
• Tests if group differences are significant

Correlation Tests

• Pearson: linear relationship between continuous variables
• Spearman: monotonic relationship (rank-based)

Confidence Intervals

• Range of values likely to contain true parameter
• 95% CI: 95% confident true value is in range
• Wider interval = less precision, higher confidence

Practice Quizzes

Quiz 1: What percentage of data falls within 2 standard deviations in a normal distribution?

• 68%
• [✓] 95%
• 99.7%
• 100%

Quiz 2: When would you use a binomial distribution?

• For continuous data
• For counting events in time
• [✓] For modeling success/failure trials
• For time between events

Quiz 3: What is hypothesis testing used for?

• Describing data
• [✓] Making decisions about population based on sample
• Calculating probabilities
• Visualizing distributions