Lecture Series on A Priori Procedure on Skew-Normal Distributions

Lecture 1: The Basics of Sampling Precision

Understanding the A Priori Procedure
Based on “From a Sampling Precision Perspective, Skewness Is a Friend and Not an Enemy!”

Welcome to Lecture 1!

Today we start an exciting journey into how we can plan our research better before collecting data. This might sound backwards, but it will make perfect sense soon!

1. What's Wrong with Traditional Statistics?

The Usual Approach

1. Collect data first
2. Run statistical tests
3. Look at p-values
4. Decide if results are "significant"

Problem: You make decisions after seeing the results! That leads to spurious significance.

A Better Way: Planning Ahead

What if we could plan our study so well that we trust our results before collecting data? That’s what the a priori procedure allows!

2. Introducing the A Priori Procedure

What Does "A Priori" Mean?

• Latin: “from what comes before”
• In statistics: make decisions before data collection
• Opposite of “a posteriori” (after seeing data)

The Three Simple Steps

1. Before collecting data, decide:
   • Desired closeness of sample mean to population mean (precision)
   • How confident you want to be about that (confidence level)
2. Calculate the minimum sample size
3. Collect data and trust your estimate!

3. Understanding the Key Concepts

Precision (f)

• How close you want your sample mean to the population mean
• Measured as a fraction of the standard deviation
• Examples:
  • \(f = 0.1\): within 0.1 SD
  • \(f = 0.3\): within 0.3 SD
  • \(f = 0.5\): within 0.5 SD
• Smaller \(f\) → more precision → closer estimate

Confidence Level

• How sure you want to be about achieving precision
• Common levels: 90%, 95%, 99%
• Corresponding \(z_c\):
  • 90% → \(z_c = 1.645\)
  • 95% → \(z_c = 1.96\)
  • 99% → \(z_c = 2.576\)

4. The Magic Formula

Basic Equation

For normally distributed data:

\[ n = \left( \frac{z_c}{f} \right)^2 \]

Where:

• \(n\): sample size
• \(z_c\): critical z-score
• \(f\): desired precision

Why It Makes Sense

• Higher confidence (larger \(z_c\)) → larger \(n\)
• Smaller \(f\) → larger \(n\)
• Quadratic relationship!

5. Real Example: Planning a Study

Scenario

• 95% confidence → \(z_c = 1.96\)
• Precision \(f = 0.3\)

Step-by-Step

\[ n = \left( \frac{1.96}{0.3} \right)^2 = (6.533)^2 = 42.68 \]

Round up → \(n = 43\)

⇒ You need 43 participants for 95% confidence that the sample mean is within 0.3 SD of the true mean.

6. Why This Approach Is Powerful

Traditional Approach	A Priori Approach
Collect data first	Plan first, then collect
Test after results	Trust results due to planning
Focus on significance	Focus on precision
Chance-driven	Robust and reliable

Key Benefits

• Transparent
• Efficient
• Trustworthy
• Flexible

7. Practice Problems

Problem 1

90% confidence (\(z_c = 1.645\)), precision \(f = 0.5\):

\[ n = \left( \frac{1.645}{0.5} \right)^2 = (3.29)^2 = 10.82 \approx 11 \]

Problem 2

99% confidence (\(z_c = 2.576\)), precision \(f = 0.2\):

\[ n = \left( \frac{2.576}{0.2} \right)^2 = (12.88)^2 = 165.89 \approx 166 \]

8. Summary: Key Takeaways

• A priori procedure = planning before data collection
• Define precision and confidence
• Use formula \(n = (z_c / f)^2\)
• Transparent and reliable
• Shift from significance testing → estimation precision

9. Looking Ahead

Next lecture: “What happens when data isn’t normal?” You’ll learn why skewness might actually help!

Homework

1. Find \(n\) for 95% confidence, \(f = 0.4\)
2. Find \(n\) for 90% confidence, \(f = 0.25\)
3. If \(n=50\), find achievable \(f\) with 95% confidence (\(\text{Hint: rearrange } f = z_c / \sqrt{n}\)).

Great job today! See you next lecture!

Lecture 2: The Problem with Normal Assumptions

Real Data Isn't Always Well-Behaved
Based on “From a Sampling Precision Perspective, Skewness Is a Friend and Not an Enemy!”

Welcome Back to Lecture 2!

Last time, we learned about the a priori procedure and how to plan studies before collecting data. Today we tackle a big question: What happens when our data isn’t perfectly normal? Spoiler: real data is messy — and that’s okay!

Quick Review: The A Priori Procedure

Our central sample size formula:

\[ n = \left( \frac{z_c}{f} \right)^2 \]

• \(n\): sample size needed
• \(z_c\): critical z-score (e.g., 1.96 for 95% confidence)
• \(f\): desired precision

This formula is elegant — but it assumes a normal distribution. What if data aren’t normal?

The Myth of the Perfect Bell Curve

What Statistics Textbooks Show

• Perfectly symmetrical
• Smooth and predictable
• “Well-behaved”

What Real Data Actually Looks Like

• Skewed: long tail on one side
• Bumpy: not smooth like textbook curves
• Irregular: outliers and odd patterns

Skewed Data is Everywhere!

Examples from Psychology

• Reaction times: many fast responses, a few very slow ones
• Confidence ratings: many “very confident,” few “not confident”
• Detection tasks: successes cluster near extremes

Examples from Everyday Life

• Income: most are moderate, a few are very high
• Social media likes: most posts have few, some go viral
• Test scores: can cluster high or low depending on difficulty

Research Evidence

Finding: Skewed distributions are the rule, not the exception.
Perfect normality is surprisingly rare in real-world data.

The Traditional Approach: Fix the “Problem”

What Researchers Are Taught

“Your data is skewed — transform it to make it normal before testing.”

Common Data Transformations

Transformation	When to Use
Square root: \(X' = \sqrt{X}\)	Moderate positive skew
Natural log: \(X' = \ln(X)\)	Strong positive skew
Log base 10: \(X' = \log_{10}(X)\)	Strong positive skew
Inverse: \(X' = \frac{1}{X}\)	Very strong positive skew
Square: \(X' = X^2\)	Negative skew (less common)

Why Transform Data?

• Meet assumptions for parametric tests (t, ANOVA)
• Improve power by stabilizing variance
• Handle outliers
• Clarify patterns in relationships

Transformation in Action

Example: Reaction Time Data

Original (ms)	Square Root Transformed
100	10.0
121	11.0
144	12.0
400	20.0
900	30.0
2500	50.0

The transformation compresses large values more than small ones, reducing skew.

The Paper’s Radical Idea

A Different Perspective

From a sampling precision perspective, skewness is a friend, not an enemy.

The Core Argument

• Traditional view: skewness ⟶ bad for tests
• New view: skewness ⟶ can help estimation precision
• It depends on your goal: testing vs estimation

Thinking About Different Research Goals

Goal 1: Significance Testing

• Prefer normality; transformations often helpful
• Skewness is typically problematic

Goal 2: Estimation Precision

• Want sample statistics close to population parameters
• A priori planning helps
• Skewness may improve precision in many practical cases

Let’s Build Some Intuition

Distribution A (Normal)

• Even spread
• More variability throughout
• Values can land anywhere

Distribution B (Skewed)

• Concentrated mass
• Smaller variability for most observations
• Many values similar

Estimation angle: When most observations cluster, the mean can be estimated precisely with fewer subjects.

Real Research Examples

Example 1: Confidence in Condom Use

• 65% “extremely confident”
• Few “not confident”
• Heavily skewed toward high confidence

Example 2: Seat Belt Attitudes

• Attitudes skewed positive across conditions
• Few negative responses

Example 3: Tanning Attitudes

• Consistently skewed
• Most responses clustered on one side

The Big Reveal

1. Technical: Equations extend the a priori procedure to skewed distributions.
2. Conceptual: Skewness can improve sampling precision.

Key Takeaways

• Skewed data are common; perfect normality is rare.
• Traditional: transform to meet test assumptions.
• New perspective: skewness may help estimation precision.
• Match methods to goals: testing vs estimation.
• Next: friendly intro to Skew-Normal distributions.

Discussion Questions

1. Other examples of skewed data in your field?
2. Why teach so much normality when reality is skewed?
3. If skewness helps precision, why the late recognition?
4. When would you not transform skewed data?

Appendix: Common Transformation Formulas

• Square root: \(X' = \sqrt{X}\)
• Natural logarithm: \(X' = \ln(X)\)
• Base-10 logarithm: \(X' = \log_{10}(X)\)
• Shifted logarithm: \(X' = \log_{10}(X + k)\) (with constant \(k\))
• Reflected square root: \(X' = \sqrt{k - X}\)
• Reflected logarithm: \(X' = \log_{10}(k - X)\) (for \(k > X\))

Great thinking today! See you next time!

Lecture 3: Introducing Skew-Normal Distributions

The Mathematical Tools for Understanding Skewed Data
Based on “From a Sampling Precision Perspective, Skewness Is a Friend and Not an Enemy!”

Welcome to Lecture 3!

Last time, we saw that real data are often skewed. Today we’ll learn the mathematical tools to work with skewness properly — step by step and friendly!

Quick Recap

• Lecture 1: A priori procedure for planning studies
• Lecture 2: Real data are often skewed, not normal
• Today: Mathematical framework for skewed data

The Normal Distribution: A Quick Review

The normal distribution has two parameters:

• Mean (\(\mu\)) — center
• Standard deviation (\(\sigma\)) — spread

Normality implies symmetry; mean sits at the center. Skewed data break this symmetry.

Introducing the Skew-Normal Distribution

The Big Idea

• Can be symmetric or skewed
• Contains the normal as a special case
• Provides tools to model and analyze skewness

The Three Parameters

Parameter	What it represents
Location (\(\xi\))	Where the distribution is centered
Scale (\(\omega\))	How spread out the main mass is
Shape (\(\lambda\))	Direction and amount of skewness

Understanding the Shape Parameter (\(\lambda\))

• \(\lambda = 0\): no skew (normal)
• \(\lambda > 0\): positive skew (right tail)
• \(\lambda < 0\): negative skew (left tail)
• Larger \(|\lambda|\): stronger skew

The Probability Density Function (PDF)

The skew-normal PDF is

\[ f(x) = \frac{2}{\omega}\,\phi\!\left(\frac{x-\xi}{\omega}\right)\, \Phi\!\left(\lambda\frac{x-\xi}{\omega}\right), \]

where \(\phi\) and \(\Phi\) are the standard normal density and CDF.

Intuition

• Start with a normal density (\(\phi\))
• “Tilt” it via the CDF factor \(\Phi(\lambda\cdot)\)
• \(\lambda\) controls the tilt/skew; \(\lambda=0\) yields the normal

Visualizing Skew-Normal Distributions

• Solid: \(\lambda=0\) (normal)
• Dotted: \(\lambda=2.0\) (moderate positive skew)
• Star: \(\lambda=-3.0\) (strong negative skew)
• Cube: \(\lambda=5.0\) (very strong positive skew)

As \(|\lambda|\) increases, the hump shifts and the curve becomes taller/narrower on the main mass side.

Key Mathematical Properties

Mean and Variance

Let \(\delta = \dfrac{\lambda}{\sqrt{1+\lambda^2}}\). Then

\[ \operatorname{Mean} = \xi + \sqrt{\frac{2}{\pi}}\,\delta\,\omega, \qquad \operatorname{Var} = \omega^2\!\left(1 - \frac{2}{\pi}\delta^2\right). \]

Unlike the normal, the mean is generally not equal to the location \(\xi\) when skewed.

Special Case: \(\lambda=0\)

• \(\delta=0\)
• Mean \(= \xi\)
• Variance \(= \omega^2\)

Important: The normal family is a subset of the skew-normal family (at \(\lambda=0\)).

Why Location/Scale Beat Mean/SD in Skewed Data

• Mean is pulled toward the tail
• Location tracks where the bulk of data sits
• Scale reflects core spread better than SD when tails are heavy

Practical Example (Income)

• Location \(\xi=\$45{,}000\)
• Scale \(\omega=\$15{,}000\)
• Shape \(\lambda=2.0\)

Compute \(\delta\): \(\displaystyle \delta=\frac{2}{\sqrt{1+4}}=\frac{2}{\sqrt{5}}\approx 0.894\).

Mean:

\[ \text{Mean} \approx 45{,}000 + \sqrt{\frac{2}{\pi}}\,(0.894)\,(15{,}000) \approx 45{,}000 + 10{,}700 \approx \$55{,}700. \]

The mean exceeds the location due to positive skew.

Why This Matters for Sampling Precision

• Greater skew (often) concentrates mass, narrowing the core spread
• Sample values cluster more tightly around the location
• \(\Rightarrow\) Potentially better precision with fewer observations

Practice: Identify Parameters

Problem 1

\(\xi=100,\ \omega=15,\ \lambda=0\).

• Distribution type?
• Mean?
• Standard deviation?

Answer: Normal; mean \(=100\); SD \(=15\).

Problem 2

\(\xi=50,\ \omega=10,\ \lambda=3.0\).

• Skew direction?
• Is mean \(>,\ =,\ <\) 50?
• Are most points nearer mean or location?

Answer: Positive skew; mean \(>50\); most points near the location (50).

Summary

• Skew-normal has location \((\xi)\), scale \((\omega)\), shape \((\lambda)\)
• \(\lambda=0\) recovers the normal
• Mean \(\neq \xi\) when skewed; \(\delta\) links \(\lambda\) to mean/variance
• Location & scale are often more informative than mean & SD under skew
• More skew can mean tighter clustering → better precision

Looking Ahead

Next time: How much does skewness improve precision, quantitatively? And how to use this in planning.

Discussion Questions

1. Why did it take so long to formalize tools for skewed data?
2. When is location more meaningful than the mean?
3. When do we need both mean and location?
4. How does this framework change your view of “problematic” skew?

Optional Deep Dive: Under the Hood

Recall the PDF:

\[ f(x) = \frac{2}{\omega}\,\phi\!\left(\frac{x-\xi}{\omega}\right)\, \Phi\!\left(\lambda\frac{x-\xi}{\omega}\right). \]

• \(\phi\big((x-\xi)/\omega\big)\): baseline normal density
• \(\Phi\big(\lambda (x-\xi)/\omega\big)\): weights one side more → skew
• \(2/\omega\): normalizes total probability to 1

Great work today! You’ve added a powerful tool to your stats toolkit.

Lecture 4: How Skewness Improves Precision

The Exciting Results and What They Mean for Research
Based on “From a Sampling Precision Perspective, Skewness Is a Friend and Not an Enemy!”

Welcome to Lecture 4!

We’ve learned the a priori procedure, saw that real data are often skewed, and added the skew-normal toolkit. Today: How much does skewness improve precision?

Quick Recap: The Journey So Far

• Lecture 1: A priori planning
• Lecture 2: Real data are skewed
• Lecture 3: Skew-normal framework
• Today: Quantifying the precision boost

The Research Question

How does skewness affect the sample size needed for a target precision and confidence?

Equivalently: with skewed vs normal data, do we need more or fewer participants for the same estimation goal?

The Experimental Setup

What They Tested

Factor	Levels Tested
Precision \(f\)	0.1, 0.2, 0.3, 0.4, 0.5
Confidence Level	90%, 95%
Skewness \(\lambda\)	0, 0.5, 1.0, 2.0, 5.0

Interpretation Aids

• \(f=0.1\): location within 0.1 scale units
• \(\lambda=0\): normal (no skew)
• \(\lambda=5.0\): very skewed

The Big Reveal

As skewness increases, the sample size needed for a given precision and confidence decreases.

This flips a common intuition from significance-testing culture.

Let’s Look at the Numbers

High Precision Example (\(f = 0.1\), 95% confidence)

Skewness \(\lambda\)	Sample Size \(n\)	Reduction
0 (Normal)	385	—
0.5	158	59% fewer
1.0	146	62% fewer
2.0	140	64% fewer
5.0	138	64% fewer

• Normal assumption: 385 participants
• Slight skew (\(\lambda=0.5\)): 158 participants — 227 fewer!
• Biggest drop from \(\lambda=0\) → \(\lambda\approx 0.5\)

Medium Precision Example (\(f = 0.3\), 95% confidence)

Skewness \(\lambda\)	Sample Size \(n\)	Reduction
0 (Normal)	43	—
0.5	22	49% fewer
1.0	19	56% fewer
2.0	18	58% fewer
5.0	17	60% fewer

• Normal: 43 participants
• Slight skew: 22 — 21 fewer

Visualizing the Results

• Curves slope downward: more skew ⇒ smaller \(n\)
• Higher-precision curves drop faster
• Big early gains from \(\lambda=0\) → \(\lambda \approx 0.5\)

The Interaction Effect

Interaction: the benefit of skewness depends on target precision.

• Low precision (\(f=0.5\)): small savings
• High precision (\(f=0.1\)): very large savings

The more precise you aim to be, the more skewness helps.

Why Does This Happen? Intuition

• Skewed distributions often concentrate mass ⇒ narrower core spread
• Samples cluster more tightly around the location
• ⇒ Better precision with fewer observations

Practical Implications

Good News for Researchers

• Save time/money: fewer participants
• Ethical: expose fewer people
• Easier recruitment
• Better precision at fixed \(n\)

Important Caveat

The precision advantage pertains to estimating the location parameter, not necessarily the mean.

Real-World Impact

Example 1: Psychology Lab (High precision, \(f=0.1\), 95%)

• Assume normal: \(n=385\)
• Typical skew (\(\lambda=1.0\)): \(n\approx 146\)
• Savings: 239 participants

Example 2: Education Research (Moderate precision, \(f=0.3\), 90%)

• Assume normal: \(n=19\)
• Typical skew (\(\lambda=1.0\)): \(n\approx 13\)
• Savings: 6 participants

Let’s Practice

Practice Problem 1

Planned \(n=100\) under normality. If \(\lambda=1.0\), how many might suffice?

Answer: Roughly 40–50% fewer ⇒ \(n\approx 50\text{–}60\).

Practice Problem 2

Why does skew help more at high precision?

Answer: Tighter clustering in skewed data helps most when targeting very fine accuracy.

Key Takeaways

• Skewness can reduce required sample sizes dramatically
• Benefits often exceed 50% fewer participants
• Biggest gains occur moving off normality (\(\lambda\approx 0 \to 0.5\))
• Interaction: the higher the precision target, the larger the gain
• Mechanism: narrower core spread ⇒ tighter sampling

Looking Ahead

Next: Applying these methods to real data—a worked example from start to finish.

Optional: The Mathematical Challenge

The authors solve for the minimum \(n\) numerically via

\[ \int_{L(n)}^{U(n)} 2\,\phi(z)\,\Phi\!\big(\sqrt{n}\,\lambda\, z\big)\,dz \;=\; c, \]

• \(L(n), U(n)\): functions of \(n\) and precision \(f\)
• \(\phi, \Phi\): standard normal PDF and CDF
• \(c\): target confidence (e.g., 0.90 or 0.95)

Amazing discoveries today — “problem” data can be super helpful!

Lecture 5: Practical Application and Examples

How to Actually Use These Methods with Real Data
Based on “From a Sampling Precision Perspective, Skewness Is a Friend and Not an Enemy!”

Welcome to Lecture 5!

We’ve seen the theory; now let’s apply it. Today we’ll take the skew-normal toolkit and walk through a complete, practical example you can replicate.

Quick Recap: What We Know

• Lectures 1–2: A priori procedure; normal assumptions often fail
• Lecture 3: Skew-normal distributions and parameters
• Lecture 4: Skewness can dramatically reduce required sample sizes
• Today: End-to-end application with real data

The Complete Process: Step by Step

1. Collect (or obtain) data
2. Estimate skewness \(\lambda\)
3. Estimate location \((\xi)\) and scale \((\omega)\)
4. Use these to plan \(n\) or construct CIs for \(\xi\)
5. Interpret correctly (location vs mean)

Real Data Example: Leaf Area Index (LAI)

Robinia pseudoacacia (black locust) LAI measured June–October 2010 (China).

What is Leaf Area Index?

• Leaf surface per ground area (0 to 6+)
• Important for growth, carbon cycling, ecosystem health

First Rows (illustrative)

June	July	September	October
4.87	3.32	2.05	1.50
5.00	3.02	2.12	1.46
4.72	3.28	2.24	1.55
5.16	3.63	2.56	1.27
5.11	3.68	2.67	1.26

Step 1: Basic Statistics

• Sample size: \(n=96\)
• Sample mean: \(\bar{x}=2.6358\)
• Sample SD: \(s=1.2099\)

These don’t reveal skewness yet.

Step 2: Estimate Skewness

Sample skewness:

\[ Sk_3 = \frac{1}{n}\sum_{i=1}^n \frac{(x_i-\bar{x})^3}{s^3} \quad \Rightarrow \quad Sk_3=0.6135. \]

• Positive \(Sk_3\) ⇒ right tail; mass concentrated on the left

Step 3: Estimate \(\lambda\) (via \(\delta\))

1. Find \(\hat{\delta}\) from \(Sk_3\)
2. Convert \(\hat{\delta}\) to \(\hat{\lambda}\)

3a. Estimate \(\delta\)

\[ \hat{\delta} = \sqrt{\frac{\pi}{2}\cdot \frac{Sk_3^2}{\,Sk_3^2 + \left(\frac{4-\pi}{2}\right)^{2/3}}} \quad \Rightarrow \quad \hat{\delta}=0.9373. \]

3b. Estimate \(\lambda\)

\[ \hat{\lambda} = \frac{\hat{\delta}}{\sqrt{1-\hat{\delta}^2}} \quad \Rightarrow \quad \hat{\lambda}=2.6888. \]

• Substantial positive skew ⇒ strong precision advantages likely

Step 4: Estimate Location and Scale

Scale (\(\omega\))

\[ \hat{\omega} = \frac{s}{\sqrt{1-\frac{2}{\pi}\hat{\delta}^2}} \quad \Rightarrow \quad \hat{\omega}=1.8224. \]

Location (\(\xi\))

\[ \hat{\xi} = \bar{x} - \sqrt{\frac{2}{\pi}}\;\hat{\delta}\;\hat{\omega} \quad \Rightarrow \quad \hat{\xi}=1.2729. \]

Step 5: Plan a New Study

• Target precision: \(f=0.1\) (within 0.1 scale units)
• Confidence: 92%
• Expected skewness: \(\lambda \approx 2.6888\)

Required Sample Size

Using the paper’s numerical method / code:

\[ n = 95. \]

Compare to Normal Assumption

• Normal, 95%: \(n=385\)
• Normal, 92%: still \(n>300\)
• Accounting for skewness: \(n=95\)

Step 6: Confidence Interval (with 96 Cases)

• Take 95 cases (≥ required \(n\))
• Sample mean (subset): \(\bar{x}_s=2.6366\)
• 92% CI for \(\xi\): \([2.3053,\;2.6424]\)
• Point estimate \(\hat{\xi}=2.4868\) falls inside

Visualization

• Histogram of data
• Fitted skew-normal curve
• Location \(\xi\) marked

Practice Example: Reaction Times

Times (ms): 210, 195, 230, 415, 188, 205, 198, 680, 215, 190

Step-by-Step

1. Basic stats: \(n=10,\ \bar{x}=272.2,\ s=152.7\)
2. Skewness: compute \(Sk_3\) from the data
3. Parameters: derive \(\hat{\delta}, \hat{\lambda}, \hat{\omega}, \hat{\xi}\)

Using R for Calculations

Numerical integration and parameter mapping are easier with software. The paper provides R utilities to compute \(n\), parameters, CIs, and plots.

R Code Skeleton

# Load the sn package for skew-normal distributions
library(sn)

# Your data
data <- c(4.87, 3.32, 2.05, 1.50, 5.00, ...)

# Basic statistics
n <- length(data)
m <- mean(data)
s <- sd(data)

# Calculate sample skewness (e.g., from e1071::skewness or custom)
# skewness <- e1071::skewness(data, type = 1)
# or compute manually:
# skewness <- mean(((data - m)^3) / s^3)

# Estimate delta and lambda
delta  <- sqrt((pi/2) * (skewness^2) / (skewness^2 + ((4 - pi)/2)^(2/3)))
lambda <- delta / sqrt(1 - delta^2)

# Estimate omega and xi
omega <- s / sqrt(1 - (2/pi) * delta^2)
xi    <- m - sqrt(2/pi) * delta * omega

Important Caveats and Considerations

When These Methods Shine

• Goal is estimation precision (not hypothesis testing)
• Data are skewed (or expected to be)
• Planning efficient studies
• Willing to use location rather than mean

When to Be Cautious

• Primary goal is significance testing / legacy comparability on means
• Very small samples
• Extremely unstable skew

Practical Tips

Planning New Studies

1. Use pilot data to estimate skew
2. Borrow plausible \(\lambda\) from literature
3. Be conservative if uncertain (smaller \(\lambda\))
4. Document \(\lambda\), \(\xi\), \(\omega\), and computations

Analyzing Existing Data

1. Check skewness first
2. Compute both normal- and skew-normal–based results
3. Report both; discuss differences
4. Interpret carefully: location \(\neq\) mean under skew

Key Takeaways

• Estimate \(\lambda\) from sample skewness, then \(\delta \to \lambda \to \omega,\ \xi\)
• Substantial savings are possible (e.g., \(n=95\) vs \(300+\))
• Software is essential for robust computation
• Interpretation hinges on location vs mean

Looking Ahead

Final lecture: the big picture—how these results should reshape statistical thinking and research design, and navigating the precision vs significance trade-off.

Excellent work today! You now have practical tools for your research.