STAT 17: Statistical Methods for Business and Economics

Prof. Marcela Alfaro Cordoba

Statistics - UCSC

29 May 2026

Case Study: Understanding Student Success

Meet Maria, a UCSC student working part-time while pursuing her degree.

Her challenge: She commutes from San Jose and wants to understand:

  • How much time should she budget for her daily commute?
  • What GPA range is typical for students in her program?
  • How many study hours do successful students put in weekly?
  • What salary can she expect after graduation?

The pattern: All these questions involve measurements that follow a bell curve - the Normal Distribution!

Understanding the normal distribution helps Maria (and you!) make informed decisions about time management, goal-setting, and career planning.

Quick Review: Continuous Random Variables

What we learned last time:

  • Continuous variables take ANY value in an interval
  • P(X = exact value) = 0 for continuous distributions
  • We calculate probabilities over intervals: P(a ≤ X ≤ b)
  • PDF (Probability Density Function) - area under curve = probability
  • P(X ≤ a) = P(X < a) for continuous variables

Key Difference from Discrete:

  • Discrete: countable values, use PMF
  • Continuous: uncountable values, use PDF and areas

Quick Review: The Uniform Distribution

X ~ Uniform(a, b): All values equally likely

PDF: f(x) = 1/(b-a) for a ≤ x ≤ b

Properties:

  • Mean: E(X) = (a + b)/2
  • Variance: Var(X) = (b-a)²/12
  • P(c ≤ X ≤ d) = (d-c)/(b-a)

Example: Bus arrives Uniform(0, 20) minutes

  • Flat probability across interval
  • Simple rectangular areas

What We’ll Accomplish Today

By the end of this lecture, you will be able to:

  • Understand properties of the normal distribution
  • Use z-score transformations to standardize values
  • Calculate probabilities using the standard normal distribution
  • Apply normal distribution to solve real-world problems using Google Sheets and/or calculators
  • Interpret results in practical contexts

The Normal Distribution: Why It Matters

The most important distribution in statistics!

Where do we see it?

  • Heights and weights of people
  • Test scores (SAT, GRE, exams)
  • Measurement errors
  • Commute times
  • Blood pressure readings
  • Product dimensions in manufacturing
  • Income distributions (log-transformed)

Why so common? Central Limit Theorem (coming soon!)

The Normal Distribution: Shape and Properties

Notation: X ~ Normal(μ, σ²) or X ~ N(μ, σ²)

Key Parameters:

  • μ (mu) = mean (center of distribution)
  • σ² (sigma squared) = variance
  • σ (sigma) = standard deviation (spread)

{Normal Distribution }

Properties:

  1. Bell-shaped and symmetric around μ
  2. Mean = Median = Mode = μ
  3. Total area under curve = 1
  4. Extends from -∞ to +∞ (but 99.7% within μ ± 3σ)
  5. Completely determined by μ and σ

The Empirical Rule (68-95-99.7 Rule)

For ANY normal distribution:

  • 68% of data within μ ± σ
  • 95% of data within μ ± 2σ
  • 99.7% of data within μ ± 3σ

The Empirical Rule (68-95-99.7 Rule)

Example: Commute time X ~ N(45, 100) minutes

  • μ = 45 minutes, σ = 10 minutes

  • 68% of commutes: 35 to 55 minutes

  • 95% of commutes: 25 to 65 minutes

  • 99.7% of commutes: 15 to 75 minutes

This helps Maria plan - she should budget 65 minutes to be 97.5% confident!

Why We Can’t Calculate by Hand

The normal PDF is complex:

\[f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\]

Problems:

  • We can’t solve an integral using this function
  • No simple formula for P(X ≤ x)

Solution: Use technology (Google Sheets, calculator, other) or standardize with z-scores!

The Standard Normal Distribution

Definition: Z ~ N(0, 1)

  • Mean μ = 0
  • Standard deviation σ = 1
  • This is our reference distribution!

Why standardize?

Instead of infinite tables for every μ and σ, we use ONE standard table and transform any normal variable to match it.

The transformation: z = (x - μ)/σ

This is called a z-score!

Understanding Z-Scores

Z-score formula: z = (x - μ)/σ

Interpretation: Number of standard deviations from the mean

Properties:

  • z = 0: exactly at the mean
  • z > 0: above the mean
  • z < 0: below the mean
  • z = 1: one SD above mean
  • z = -2: two SDs below mean

Example: Maria’s commute X ~ N(45, 100)

  • Today’s commute: 60 minutes

  • z = (60 - 45)/10 = 1.5

  • Interpretation: 1.5 standard deviations above average (slower than usual!)

Z-Score Practice

Heights of UCSC students: X ~ N(66, 9) inches

Calculate and interpret z-scores:

  1. Student A: 69 inches tall
    • z = (69 - 66)/3 = 1
    • One SD above average
  2. Student B: 60 inches tall
    • z = (60 - 66)/3 = -2
    • Two SDs below average

THINK-PAIR-SHARE 1 (7 minutes)

Understanding Z-Scores

GPA at UCSC: X ~ N(3.2, 0.16), so σ = 0.4

Calculate z-scores and interpret:

  1. Maria’s GPA: 3.6
  2. Her roommate’s GPA: 2.8
  3. Dean’s List requirement: 3.5
  4. Who is further from the mean (in SDs)?

Bonus: What GPA corresponds to z = -1?

Share your answers in Poll Everywhere!

Answer question 4: Who is further from the mean?

Using Google Sheets for Normal Probabilities

Two essential functions:

=NORM.DIST(x, mean, std_dev, TRUE)

Returns P(X ≤ x) : cumulative probability

=NORM.INV(probability, mean, std_dev)

Returns x-value for given cumulative probability

For Standard Normal (Z):

=NORM.S.DIST(z, TRUE)  
=NORM.S.INV(probability)

Example: Finding P(X ≤ x)

Maria’s commute: X ~ N(45, 100) minutes

Question: What’s the probability her commute is 50 minutes or less?

Solution in Google Sheets:

=NORM.DIST(50, 45, 10, TRUE)

Answer: ≈ 0.6915 or 69.15%

Interpretation: About 69% of the time, Maria’s commute takes 50 minutes or less.

Question in exam Given this information, what is the probability her commute is 50 minutes or more? Draw the distribution and identify the area under the curve you are calculating with that function.

Example: Finding P(X > x)

Question: Probability Maria’s commute exceeds 55 minutes?

Remember: P(X > x) = 1 - P(X ≤ x)

Solution in Google Sheets:

=1 - NORM.DIST(55, 45, 10, TRUE)

Answer: ≈ 0.1587 or 15.87%

Interpretation: About 16% of days, she should expect a commute longer than 55 minutes.

Question in exam Given this information, what is the probability that Maria’s commute is equal or exceeds 55 minutes? Draw the distribution and identify the area under the curve you are calculating with that function.

Example: Finding P(a ≤ X ≤ b)

Question: Probability Maria’s commute is between 40 and 50 minutes?

Formula: P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a)

Solution in Google Sheets:

=NORM.DIST(50, 45, 10, TRUE) - NORM.DIST(40, 45, 10, TRUE)

Step by step:

  • P(X ≤ 50) ≈ 0.6915

  • P(X ≤ 40) ≈ 0.3085

  • P(40 ≤ X ≤ 50) ≈ 0.3830

Interpretation: About 38% of her commutes fall in this range.

Question in exam Given this information, what is the probability that Maria’s commute is more than 50 minutes? Draw the distribution and identify the area under the curve you are calculating with that function.

Example: Finding the Value (Inverse)

Question: Maria wants to leave early enough so she arrives on time 90% of days. How much time should she budget?

We need: x such that P(X ≤ x) = 0.90

Solution in Google Sheets:

=NORM.INV(0.90, 45, 10)

Answer: ≈ 57.8 minutes

Interpretation: Budget 58 minutes to be 90% confident of arriving on time!

-> How do we represent this in a plot?

All Types of Normal Probabilities

For X ~ N(μ, σ²):

Type Formula Google Sheets
P(X ≤ x) Direct =NORM.DIST(x, μ, σ, TRUE)
P(X < x) Same as ≤ =NORM.DIST(x, μ, σ, TRUE)
P(X ≥ x) 1 - P(X ≤ x) =1-NORM.DIST(x, μ, σ, TRUE)
P(X > x) Same as ≥ =1-NORM.DIST(x, μ, σ, TRUE)
P(a ≤ X ≤ b) P(X≤b)-P(X≤a) =NORM.DIST(b,μ,σ,TRUE)-NORM.DIST(a,μ,σ,TRUE)

Key: For continuous distributions, < and ≤ give the same result!

🧘‍♀️ STRETCH BREAK

Time to move! (5 minutes)

  • Stand up and stretch 🤸‍♀️
  • Chat with neighbors 💬
  • Grab some water 💧

When we return: Normal distribution applications

Real Application: Study Hours

Successful UCSC students study X ~ N(15, 16) hours per week

Scenario: Maria wants to understand study patterns.

Calculate:

  1. P(X ≤ 12) - students studying 12 hours or less

    =NORM.DIST(12, 15, 4, TRUE) ≈ 0.2266

    About 23% of students

  2. P(12 ≤ X ≤ 18) - students in the “typical” range

    =NORM.DIST(18, 15, 4, TRUE) - NORM.DIST(12, 15, 4, TRUE) ≈ 0.5468

    About 55% fall in this range

THINK-PAIR-SHARE 2 (7 minutes)

Normal Distribution Applications

Starting salary for UCSC graduates in Maria’s field: X ~ N(65000, 100000000) dollars (σ = $10,000)

Calculate using Google Sheets:

  1. P(X ≤ 60000) - earning $60k or less
  2. P(X ≥ 75000) - earning $75k or more
  3. P(60000 ≤ X ≤ 70000) - middle range
  4. What salary represents the 75th percentile?
  5. Between what two salaries do the middle 50% fall?

Bonus: If Maria wants to be in the top 10% of earners, what salary does she need?

Share your answers in Poll Everywhere!

What salary represents the 75th percentile?

Working with Standard Normal (Z)

When to use Z ~ N(0, 1):

If you’ve already calculated z-scores, use standard normal functions!

Example: z = 1.5

=NORM.S.DIST(1.5, TRUE) ≈ 0.9332

Interpretation: 93.32% of data falls below z = 1.5

Inverse:

=NORM.S.INV(0.95) ≈ 1.645

Interpretation: z = 1.645 is the 95th percentile

Converting Between X and Z

Two approaches for the same problem:

Approach 1: Work directly with X

=NORM.DIST(x, μ, σ, TRUE)

Approach 2: Convert to Z first, then use standard normal

z = (x - μ)/σ
=NORM.S.DIST(z, TRUE)

Both give the same answer! Use whichever feels more comfortable.

Tip: Approach 1 is usually faster and less prone to arithmetic errors.

Real Application: Test Scores

Midterm scores: X ~ N(75, 64), so σ = 8

Questions for the class:

  1. What percentage scored below 70?

    =NORM.DIST(70, 75, 8, TRUE) ≈ 0.2660

    About 27%

  2. Professor A curves: top 15% get A’s. What’s the cutoff?

    =NORM.INV(0.85, 75, 8) ≈ 83.29

    Need 84+ for an A

Finding Percentiles

Percentile: The value below which a percentage of data falls

Common percentiles:

  • 25th (Q1), 50th (median/Q2), 75th (Q3)
  • 90th (top 10%), 95th (top 5%)

Formula in Google Sheets:

=NORM.INV(percentile/100, mean, std_dev)

Example: 80th percentile of Maria’s commute

=NORM.INV(0.80, 45, 10) ≈ 53.4 minutes

The Interquartile Range (IQR)

IQR = Q3 - Q1 (middle 50% of data)

For normal distributions:

Q1 = =NORM.INV(0.25, μ, σ)
Q3 = =NORM.INV(0.75, μ, σ)
IQR = Q3 - Q1

Example: Study hours X ~ N(15, 16)

Q1 = =NORM.INV(0.25, 15, 4) ≈ 12.3 hours
Q3 = =NORM.INV(0.75, 15, 4) ≈ 17.7 hours
IQR ≈ 5.4 hours

Interpretation: The middle 50% of students study between 12-18 hours weekly.

THINK-PAIR-SHARE 3 (7 minutes)

Comprehensive Normal Distribution Problem

Credit hours per quarter at UCSC: X ~ N(16, 4), so σ = 2

Answer these questions:

  1. What percentage of students take fewer than 14 units?
  2. What percentage take more than 18 units?
  3. What percentage take between 15 and 17 units?
  4. How many units represents the 60th percentile?
  5. Between what two values do the middle 80% of students fall?
  6. Maria takes 19 units. What’s her z-score and what does it mean?

Use Google Sheets for all calculations!

Share your answers in Poll Everywhere!

Between what two values do the middle 80% of students fall?

Common Mistakes to Avoid

Mistake 1: Confusing σ and σ² - Parameters are (μ, σ²) but Google Sheets uses σ!

Mistake 2: Forgetting to use TRUE for cumulative

=NORM.DIST(x, μ, σ, FALSE)  ← PDF (rarely needed)
=NORM.DIST(x, μ, σ, TRUE)   ← CDF (what we want!)

Mistake 3: P(X > x) without the complement

Wrong: =NORM.DIST(x, μ, σ, TRUE)
Right: =1-NORM.DIST(x, μ, σ, TRUE)

Mistake 4: Using the wrong function for inverse problems - Finding values: use NORM.INV, not NORM.DIST

Summary: Normal Distribution Mastery

Key Concepts:

  1. Normal Distribution: Bell-shaped, symmetric, defined by μ and σ
  2. Empirical Rule: 68-95-99.7 for μ ± σ, μ ± 2σ, μ ± 3σ
  3. Z-scores: Standardize any value: z = (x - μ)/σ
  4. Google Sheets Functions:
    • NORM.DIST for probabilities
    • NORM.INV for values/percentiles
    • Standard versions: NORM.S.DIST, NORM.S.INV

Why we use Google Sheets: Integrals don’t have a close solution for the normal pdf, cannot be calculated by hand!

Applications: Test scores, commute times, salaries, measurements - the normal distribution is everywhere!

Back to Maria’s Story

Maria now understands:

✅ Budget 58 minutes for her commute (90% confidence)

✅ Her 3.6 GPA is above average (z = 1, top 16%)

✅ Studying 15-20 hours/week puts her in the successful range

The normal distribution helps her:

  • Set realistic expectations

  • Plan her schedule effectively

  • Make data-informed decisions

This is the power of understanding statistics!

Quick Knowledge Check ✅

Rate your confidence (1-5) on Ed Discussion:

  1. Understanding normal distribution properties ⭐⭐⭐⭐⭐
  2. Calculating and interpreting z-scores ⭐⭐⭐⭐⭐
  3. Using NORM.DIST for probabilities ⭐⭐⭐⭐⭐
  4. Using NORM.INV for percentiles ⭐⭐⭐⭐⭐
  5. Applying normal distribution to real problems ⭐⭐⭐⭐⭐

If you rated anything 3 or below, visit office hours!

Thank you! 📊✨

Questions? I have office hours right after class today!

Next up: More continuous distributions and the Central Limit Theorem

Remember:

  • Post Think-Pair-Share on Ed Discussion and Poll Everywhere
  • Rate your confidence
  • Practice with Google Sheets for normal distribution problems
  • Review z-score interpretations