🎯⭐ INTERACTIVE LESSON

Normal Distributions

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Normal Distributions - Complete Interactive Lesson

Part 1: The Normal Curve

📊 The Normal Distribution

Part 1 of 7 — Bell Curves and the Empirical Rule

The Normal Distribution

The normal distribution is the most important distribution in statistics. It is:

Symmetric and bell-shaped
Described by two parameters: mean $\mu$ and standard deviation $\sigma$
Notation: $X \sim N(\mu, \sigma)$

The Empirical Rule (68-95-99.7)

For any normal distribution:

Range	Percentage
$\mu \pm 1\sigma$	68% of data
$\mu \pm 2\sigma$	95% of data
$\mu \pm 3\sigma$

Example: IQ scores follow $N(100, 15)$

68% of scores between $100 \pm 15 = [85, 115]$
95% between $100 \pm 30 = [70, 130]$
99.7% between

🔑 The Empirical Rule gives a quick estimate for normal data. For exact probabilities, use z-scores.

Normal Distribution Check 🎯

Empirical Rule Calculations 🧮

Adult male heights follow $N(70, 3)$ inches.

1) What percentage of men are between 67 and 73 inches tall?

2) What percentage are shorter than 64 inches? (Hint: 64 = 70 − 2(3))

3) Between what two heights do the middle 99.7% of men fall? Give the upper bound.

Part 2: Z-Scores

📏 Z-Scores and the Standard Normal

Part 2 of 7 — Standardizing Values

The Z-Score Formula

A z-score tells you how many standard deviations a value is from the mean:

$z = \frac{x - \mu}{\sigma}$

Z-Score	Interpretation

Part 3: Normal Calculations

🔢 Normal Probability Calculations

Part 3 of 7 — Finding Areas and Percentiles

Forward Problems: X → Z → Probability

Given a value $x$ , find the probability:

Compute $z = \frac{x - \mu}{\sigma}$

Part 4: Assessing Normality

📈 Assessing Normality

Part 4 of 7 — Is the Data Normal?

Why Check Normality?

Many statistical procedures assume the data comes from a normal distribution. Before applying them:

Check with a histogram — should be roughly bell-shaped
Use a normal probability plot (Q-Q plot) — points should follow a straight line
Apply the Empirical Rule — about 68/95/99.7% should fall within 1/2/3 SDs

Normal Probability Plot (Q-Q Plot)

Pattern	Interpretation
Points follow a straight line	Data is approximately normal
Points curve up at both ends	Data has heavier tails (leptokurtic)
S-shaped curve	Data is skewed
Points curve down at both ends	Data has lighter tails (platykurtic)

🔑 No real data is perfectly normal. We look for "close enough" — roughly symmetric with no extreme outliers.

Normality Assessment 🎯

Part 5: Combining Normal RVs

➕ Combining Normal Random Variables

Part 5 of 7 — Sums, Differences, and Linear Transformations

Linear Transformations

If $X \sim N(\mu, \sigma)$ and $Y = a + bX$ , then:

Part 6: Problem-Solving Workshop

🛠️ Normal Distribution Workshop

Part 6 of 7 — Comprehensive Practice

Strategy for Normal Distribution Problems

Identify $\mu$ and $\sigma$ from the problem
Sketch the curve and shade the desired region
Standardize using $z = (x - \mu)/\sigma$

Part 7: Review & Applications

📋 Normal Distribution Review

Part 7 of 7 — Summary and Applications

Key Formulas

Formula	When to Use
$z = \frac{x - \mu}{\sigma}$	Convert any value to standard normal

100 \pm 45 = [55, 145]

The Standard Normal Distribution

When we standardize: $Z \sim N(0, 1)$

This allows us to use one table (or calculator) for all normal distributions.

Example: Heights $\sim N(70, 3)$ . A person is 76 inches tall. $z = \frac{76 - 70}{3} = 2$ They are 2 standard deviations above the mean.

The z-table gives $P(Z \leq z)$ — the area to the left of $z$ .

To Find	Method
$P(Z \leq z)$	Read directly from table
$P(Z \geq z)$	$1 - P(Z \leq z)$
$P(a \leq Z \leq b)$	$P(Z \leq b) - P(Z \leq a)$

🔑 Always sketch the normal curve, shade the region, then calculate.

Z-Score Practice 🎯

Z-Score Calculations 🧮

ACT scores follow $N(21, 5)$ .

1) Find the z-score for a student who scored 31. (Give as a whole number)

2) Find the z-score for a student who scored 16.

3) A student has $z = -0.4$ . What was their ACT score?

Look up

P(Z \leq z)

in the z-table

Adjust for the direction (left tail, right tail, between)

Example: Scores $\sim N(500, 100)$ . Find $P(X > 650)$ .

$z = \frac{650 - 500}{100} = 1.5$
$P(Z \leq 1.5) = 0.9332$
$P(X > 650) = 1 - 0.9332 = 0.0668 = 6.68\%$

Backward Problems: Probability → Z → X

Given a percentile, find the value:

Find the z-score from the table that matches the given probability
Solve for $x = \mu + z\sigma$

Example: What score is at the 90th percentile if $\mu = 500, \sigma = 100$ ?

90th percentile → $z = 1.28$ (from table: $P(Z \leq 1.28) = 0.8997 \approx 0.90$ )
$x = 500 + 1.28(100) = 628$

🔑 "Top 10%" = 90th percentile. "Bottom 25%" = 25th percentile.

Normal Calculations 🎯

Finding Percentiles 🧮

Baby weights at birth follow $N(7.5, 1.2)$ lbs.

1) What z-score corresponds to a baby weighing 9.9 lbs?

2) Using $P(Z \leq 2) = 0.9772$ , what percent of babies weigh less than 9.9 lbs? (Express as a number, e.g., 97.72)

3) The 84th percentile has $z \approx 1$ . What is the 84th percentile weight? (in lbs, one decimal)

Y \sim N(a + b\mu, |b|\sigma)

Y \sim N (a + b μ, ∣ b ∣ σ)

Example: Temperature in Celsius is $C \sim N(20, 5)$ . In Fahrenheit: $F = 32 + 1.8C$ $F \sim N(32 + 1.8(20),; 1.8(5)) = N(68, 9)$

Sum of Independent Normal RVs

If $X \sim N(mu_X, sigma_X)$ and $Y \sim N(mu_Y, sigma_Y)$ are independent:

$X + Y \sim N(mu_X + mu_Y, \sqrt{sigma_X^2 + sigma_Y^2})$

$X - Y \sim N(mu_X - mu_Y, \sqrt{sigma_X^2 + sigma_Y^2})$

⚠️ Variances add for both sums AND differences. Standard deviations do NOT add directly.

Example: Coffee fill $X \sim N(12, 0.3)$ oz, cream $Y \sim N(1, 0.1)$ oz. $X + Y \sim N(13, \sqrt{0.09 + 0.01}) = N(13, \sqrt{0.10}) \approx N(13, 0.316)$

Combining RVs 🎯

Combining Normal Variables 🧮

Package weights: $X \sim N(50, 4)$ lbs. Packing material: $Y \sim N(2, 0.5)$ lbs (independent).

1) Mean total weight $E(X+Y) =$ ?

2) Variance of total weight $\text{Var}(X+Y) = sigma_X^2 + sigma_Y^2 =$ ?

3) SD of total weight = $\sqrt{\text{Var}(X+Y)}$ , rounded to 2 decimal places.

Use the table or calculator to find probabilities

For percentiles: work backward from probability to z to x

Problem: A machine fills cereal boxes with $\mu = 368$ g and $\sigma = 4$ g. What proportion of boxes have less than 360 g?

$z = (360 - 368)/4 = -8/4 = -2$
$P(Z \leq -2) = 0.0228$
About $2.28\%$ of boxes are underfilled.

Follow-up: What weight is exceeded by 90% of boxes?

"Exceeded by 90%" means 10th percentile (10% are below)
$z = -1.28$ (from table)
$x = 368 + (-1.28)(4) = 368 - 5.12 = 362.88$ g

Workshop Problems 🎯

Common z-Values to Know

Confidence Level	z*
90%	1.645
95%	1.960
99%	2.576

Final Review 🎯

Mastery Review 🔽

Normal Distributions

The Standard Normal Distribution

Using the Z-Table

Backward Problems: Probability → Z → X

Sum of Independent Normal RVs

Worked Example

Common z-Values to Know