FAD1015 L15-L16 — Normal Distribution & Approximation

Lectures 15-16 covering the normal (Gaussian) distribution, the most important continuous distribution in statistics. Source file: (L15 L16) FAD 1015 -Week 8 Normal Distribution and Approximation_ (1).pdf

Summary

Comprehensive coverage of the normal distribution: properties, standard normal, z-scores, probability calculations using tables, and normal approximation to binomial with continuity correction. Includes foundational concepts (population vs sample, parameter vs statistic) and numerous worked examples.

Key Concepts

  • Probability Distributions — Normal distribution
  • Normal (Gaussian) Distribution — bell-shaped curve
  • Standard Normal Distribution — Z ~ N(0, 1)
  • Z-Score — standardization: Z = (X - μ)/σ
  • Normal Approximation to Binomial
  • Continuity Correction
  • Central Limit Theorem (introduced)

Lecture Coverage

1. Foundations: Population vs Sample & Parameter vs Statistic

Population: The whole set of items that are of interest. Sample: Some subset of the population intended to represent the population.

Parameter: A number that describes the data from a population. Statistic: A number that describes the data from a sample.

Measure Population Parameter Sample Statistic
Mean μ X̄
Variance σ² s²
Standard Deviation σ s

2. Continuous Probability Distribution Recap

  • Distribution of a continuous random variable (e.g., heights, weights, amount of milk)
  • Total probability of all intervals = 1
  • Probability in any interval lies in range [0, 1]
  • Represented by a density curve

3. Normal Distribution Properties

  • Bell-shaped, symmetric about mean
  • Unimodal (single peak)
  • Mean = Median = Mode
  • Asymptotic to x-axis
  • Defined by mean (μ) and standard deviation (σ)
  • Changing μ shifts the curve left/right
  • Changing σ changes the spread (larger σ = wider/flatter)

4. Probability Density Function

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(\frac{-(x-\mu)^2}{2\sigma^2}\right), \quad -\infty < x < +\infty$$

Notation: X ~ N(μ, σ²)

The total area under the normal curve = 1.

5. Empirical Rule (68-95-99.7 Rule)

  • About 68% of data falls between μ − σ and μ + σ
  • About 95% of data falls between μ − 2σ and μ + 2σ
  • About 99.7% of data falls between μ − 3σ and μ + 3σ

More precisely:

  • Area between μ ± σ: 0.6827
  • Area between μ ± 2σ: 0.9545
  • Area between μ ± 3σ: 0.9973

6. Standard Normal Distribution

The normal distribution with μ = 0 and σ = 1 is called the standard normal distribution.

Standardization: Any X ~ N(μ, σ²) can be transformed to Z ~ N(0, 1) using:

$$Z = \frac{X - \mu}{\sigma}$$

Z-values (Z scores) represent the distance between a selected value X and the population mean μ, divided by the population standard deviation σ.

Symmetry properties:

  • P(Z < 0) = P(Z > 0) = 0.5
  • P(Z < −z) = P(Z > z)
  • P(Z > −z) = 1 − P(Z > z)

7. Using the Normal Distribution Table

The standard normal table gives P(Z > z) — the area in the right tail.

To find probabilities:

  • P(Z > a): read directly from table
  • P(Z < a) = 1 − P(Z > a)
  • P(a < Z < b) = P(Z > a) − P(Z > b)
  • P(|Z| < a) = 1 − 2·P(Z > a)

8. Probability Calculations (General Normal)

graph LR
    START["Given X ~ N(mu, sigma^2)"] --> STEP1["Step 1: Standardize<br/>Z = (X - mu) / sigma"]
    STEP1 --> STEP2["Step 2: Use standard<br/>normal table"]
    STEP2 --> STEP3["Step 3: Interpret result"]

    style START fill:#e7f5ff,stroke:#1971c2
    style STEP1 fill:#ffe8cc,stroke:#d9480f
    style STEP2 fill:#e5dbff,stroke:#5f3dc4
    style STEP3 fill:#d3f9d8,stroke:#2f9e44

For X ~ N(μ, σ²):

  1. Standardize: Z = (X − μ)/σ
  2. Use standard normal table to find probability
  3. Interpret result

Types of problems:

  • Find P(X > a), P(X < a), P(a < X < b)
  • Find the value of a given P(X < a) = p
  • Find μ or σ given other information

9. Normal Approximation to Binomial

graph TD
    START["X ~ B(n,p)"] --> CHECK{"np > 5<br/>AND<br/>nq > 5?"}
    CHECK -->|No| WARN["Normal approximation<br/>NOT appropriate"]
    CHECK -->|Yes| CALC["Calculate:<br/>mu = np<br/>sigma^2 = npq"]
    CALC --> CC["Apply continuity correction"]
    CC --> STD["Standardize:<br/>Z = (Y - mu) / sigma"]
    STD --> TABLE["Use standard normal table"]

    style START fill:#e7f5ff,stroke:#1971c2
    style CHECK fill:#fff4e6,stroke:#e67700
    style WARN fill:#ffe3e3,stroke:#c92a2a
    style CALC fill:#e5dbff,stroke:#5f3dc4
    style CC fill:#ffe8cc,stroke:#d9480f
    style STD fill:#c5f6fa,stroke:#0c8599
    style TABLE fill:#d3f9d8,stroke:#2f9e44

For X ~ B(n, p), when np > 5 and nq > 5 (where q = 1 − p):

X can be approximated by Y ~ N(μ, σ²) where:

  • μ = np
  • σ² = npq
  • σ = √(npq)

The shape of the binomial distribution needs to be similar to the normal distribution. The normal distribution works best when p is close to 0.5 and n is large.

Common error: Accidentally forgetting to square root the variance to get the standard deviation — needed in probability calculations.

10. Continuity Correction

Approximating the binomial (discrete) using the normal (continuous) requires a continuity correction.

graph LR
    subgraph discrete["Discrete: X"]
        D1["P(X <= k)"]
        D2["P(X < k)"]
        D3["P(X >= k)"]
        D4["P(X > k)"]
        D5["P(X = k)"]
        D6["P(a <= X <= b)"]
    end
    subgraph cont["Continuous: Y"]
        C1["P(Y < k + 0.5)"]
        C2["P(Y < k - 0.5)"]
        C3["P(Y > k - 0.5)"]
        C4["P(Y > k + 0.5)"]
        C5["P(k - 0.5 < Y < k + 0.5)"]
        C6["P(a - 0.5 < Y < b + 0.5)"]
    end

    D1 --> C1
    D2 --> C2
    D3 --> C3
    D4 --> C4
    D5 --> C5
    D6 --> C6

    style discrete fill:#e7f5ff,stroke:#1971c2
    style cont fill:#d3f9d8,stroke:#2f9e44

In the discrete distribution, each probability is represented by a rectangle. When working out probabilities, we want to include whole rectangles.

Discrete Continuous Equivalent
P(X ≤ k) P(Y < k + 0.5)
P(X < k) P(Y < k − 0.5)
P(X ≥ k) P(Y > k − 0.5)
P(X > k) P(Y > k + 0.5)
P(X = k) P(k − 0.5 < Y < k + 0.5)
P(a ≤ X ≤ b) P(a − 0.5 < Y < b + 0.5)

11. Central Limit Theorem (Brief)

For non-normal distributions, as the sample size increases, the distribution of the sample means becomes approximately Normal. If the sample size is large enough, the distribution looks normally distributed.

Worked Examples from Lecture

Example 1: Empirical Rule

Test scores of 500 students: X ~ N(70, 5²)

  • P(65 < X < 75) = P(μ − σ < X < μ + σ) ≈ 68%
  • P(70 < X < 80) = P(μ < X < μ + 2σ) ≈ 34% + 13.5% = 47.5%

Example 2: Standard Normal Probabilities

If Z ~ N(0,1), find:

  • P(Z > 2.43)
  • P(Z < −2.35)
  • P(Z > −1.28)
  • P(0.5 < Z < 1.23)
  • P(−1.75 < Z < −1.25)

Example 3: General Normal Probabilities

If X ~ N(300, 25), find:

  • P(X > 305)
  • P(X < 291)
  • P(286 < X < 312)

Example 4: Finding Parameters

The lifetimes of car tyres are N(7500, σ²). Find σ if 5% last less than 6000 km.

Example 5: Normal Approximation

X ~ B(20, 0.4). Use normal approximation to estimate P(X ≤ 6).

Check: np = 8 > 5, nq = 12 > 5 ✓ μ = 8, σ² = 4.8, σ = √4.8 ≈ 2.19 P(X ≤ 6) ≈ P(Y < 6.5) = P(Z < (6.5−8)/2.19)

Example 6: Continuity Correction Practice

X ~ B(500, 0.5). Use normal approximation for:

  • P(X < 260) → P(Y < 259.5)
  • P(X ≤ 250) → P(Y < 250.5)
  • P(X > 255) → P(Y > 255.5)
  • P(X ≥ 235) → P(Y > 234.5)
  • P(X = 260) → P(259.5 < Y < 260.5)

Exercises

The lecture includes extensive exercises (14 sets) covering:

  • Basic probability calculations
  • Finding cutoff values
  • Finding μ and σ from percentiles
  • Normal approximation with continuity correction

Related Topics

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