Pairing Basics Slides The Power of Pairing
Reducing Noise, Finding Signal
Statistics
Matched Pairs
The Sunscreen Scenario
Option A: Independent
Test SPF 30 on 50 people in Florida and SPF 50 on 50 different people in Oregon.
Too much background noise!
Option B: Matched Pairs
Test SPF 30 on the left arm and SPF 50 on the right arm of the same 50 people.
Comparison is within each person!
Independent Samples
Two Separate Groups
Individuals in Group 1 have no connection to individuals in Group 2.
Structure
n₁ and n₂
Goal
Compare means
Example
Boys vs. Girls
Paired Samples (Matched Pairs)
Natural Connection
Data points are linked by a subject or a specific matching characteristic.
The "Self-Control"
Same person gets both treatments (Before/After or Left/Right).
The "Matched Match"
Two different people paired by age, IQ, twin status, or location.
Why do we pair?
In the real world, people are vastly different. These differences create "noise" (variation).
Pairing blocks out the noise.
By comparing a person to themselves, we remove all their individual quirks and focus only on the effect of the treatment.
Quick Quiz: Pair or Not?
1. A scientist measures the weight of 20 mice before and after a high-protein diet.
PAIRED
2. A researcher compares the average height of 100 men and 100 women.
INDEPENDENT
3. 50 sets of identical twins are split: one twin gets a drug, the other gets a placebo.
PAIRED
Design Detective Worksheet Design Detective
Independent Samples vs. Matched Pairs
Name:
Date:
The Key Question: Is there a meaningful link between a data point in Group 1 and a specific data point in Group 2? If yes, it's Matched Pairs . If no, they are Independent Samples .
Case Study Analysis
For each scenario below: (1) Identify the design, (2) Explain your reasoning, and (3) Identify the variable being measured.
Scenario A: A fitness tracker company wants to test if their new heart rate sensor is more accurate than the old one. They have 25 volunteers wear the new model on their left wrist and the old model on their right wrist while running on a treadmill.
Design Type
Independent
Matched Pairs
Measured Variable
Reasoning
Scenario B: To determine if a new fertilizer increases crop yield, a researcher randomly assigns 20 plots of land to receive the new fertilizer and another 20 different plots of land to receive the current standard fertilizer.
Design Type
Independent
Matched Pairs
Measured Variable
Reasoning
Scenario C: A psychologist wants to know if a specific meditation technique reduces anxiety. She measures the anxiety levels of 40 participants, teaches them the technique, and then measures their anxiety levels again after one month of practice.
Design Type
Independent
Matched Pairs
Measured Variable
Reasoning
Explain the "Why"
In Scenario A , why is it better to have the same person wear both trackers rather than having 25 people wear the new one and a different group of 25 people wear the old one? Discuss how individual differences (like arm movement or skin thickness) might affect the results.
Difference Dynamics Slides The Difference Method
Reducing Two Lists to One Truth
Lesson 2: Distribution of Differences
The Power of \(x_d\)
In a matched pairs design, we don't care about the original values. We care about the change .
\[ \text{Difference } (x_d) = \text{Post} - \text{Pre} \]
Note: You can also do Pre - Post, as long as you stay consistent!
Pre Post Diff (\(x_d\)) 120 128 +8 115 112 -3 ... ... ...
Visualizing the Shift
Dotplots & Histograms
We graph only the differences. We are looking for the shape of the change .
Boxplots
Does the median difference sit at 0? If so, the treatment might not be doing much!
Distribution of \(x_d\)
The Normality Check
Before we can use a t-test on our differences, we must verify the Normal/Large Sample condition.
Scenario 1
Population is Normal
The differences in the population are stated to be normal.
Scenario 2
Sample Size \(n \ge 30\)
Central Limit Theorem kicks in! We are good to go.
Scenario 3
Sample is Small (\(n < 30\))
Plot the differences! Ensure no strong skew or outliers.
Reduce to One
Once you calculate the differences, ignore the original data.
You now have a single sample of data points. We are back to 1-sample t-procedures!
Difference Calculation Workshop Difference Workshop
Step 1: Reduce. Step 2: Visualize.
STUDENT:
DATE:
To analyze paired data, we calculate the difference for each pair . We then treat these differences as a single sample. In this workshop, we use \( \text{Difference} = \text{Post} - \text{Pre} \).
Activity 1: Jumping Jacks & Heart Rate
Ten students measured their heart rate (BPM) before and after doing 30 seconds of jumping jacks.
Student Pre Post Diff (\(x_d\)) 1 72 115 2 68 102 3 80 130 4 75 110 5 64 95 6 70 105 7 78 122 8 72 118 9 66 100 10 82 125
Dotplot of Differences (\(x_d\))
20
30
40
50
60
Describe the Shape
Is Normality met for \(n=10\)? Explain.
Visualizing "No Difference"
Suppose a study found that a new study app had absolutely no effect on test scores. If you calculated the differences (\( \text{New} - \text{Old} \)), what would you expect the boxplot of those differences to look like? Sketch it below.
-10
0
10
Sketch boxplot here
Key Landmark
Where should the center of the distribution be located?
Variability
Does "no effect" mean every difference must be exactly zero? Why or why not?
Inference Procedures Slides Inference for Paired Means
Testing the Average Difference
Lesson 3: Hypotheses & Intervals
The "One-Sample" Secret
Once we have our list of differences (\(x_d\)), the two original populations disappear.
Treat it like a 1-Sample t-test!
We are no longer comparing \(\mu_1\) vs \(\mu_2\). We are testing \(\mu_d\) (the true mean difference) against zero.
The Hypotheses
Null Hypothesis (\(H_0\))
\[ \mu_d = 0 \]
"There is no mean difference."
Alternative (\(H_a\))
\[ \mu_d > 0 \text{ or } \mu_d < 0 \text{ or } \mu_d \neq 0 \]
The Math of Inference
Test Statistic
\[ t = \frac{\bar{x}_d - 0}{s_d / \sqrt{n}} \]
Where \( \bar{x}_d \) is the sample mean difference and \( s_d \) is the sample standard deviation of differences.
Confidence Interval
\[ \bar{x}_d \pm t^* \left( \frac{s_d}{\sqrt{n}} \right) \]
Use degrees of freedom: \( df = n - 1 \).
Interpretation
CI
"We are 95% confident that..."
"...the true mean difference [in context] is between [lower] and [upper] units."
Does the interval contain 0?
Yes (includes 0)
We do NOT have convincing evidence of a mean difference.
No (all positive or negative)
We HAVE convincing evidence of a mean difference!
SAT Success?
"Does a prep course work?" is a paired question.
"If the interval for \(\mu_d\) (After - Before) is \([45, 85]\), we are 95% confident that the course increases scores by an average of 45 to 85 points."
Mean Difference Lab Worksheet Mean Difference Lab
Applying Paired t-Procedures
Name:
Scenario: Energy Boost?
A researcher wants to know if a new energy drink improves reaction time. Ten volunteers were tested before and after drinking 8oz of the "Spark" energy drink. Reaction times (in milliseconds) were recorded.
Summary Statistics for Differences (\( \text{After} - \text{Before} \))
n
10
\(\bar{x}_d\)
-12.4 ms
\(s_d\)
7.2 ms
Step 1: State Hypotheses
\( H_0: \mu_d \)
\( H_a: \mu_d \)
Parameter Definition
Define \(\mu_d\) in context:
Step 2: Calculate Test Statistic (\(t\)) & P-value
\( t = \)
\( P\text{-value} = \)
\( df = \)
Step 3: Conclude (at \(\alpha = 0.05\))
Step 4: Confidence Interval
Construct a 95% confidence interval for the true mean difference.
Taste Test Tracker Taste Test Tracker
Live Data Collection Protocol
Class Section:
The Experiment
Does students' preference for Brand A vs. Brand B differ on a 1-10 scale? Each participant tastes both samples (blinded) and provides a rating.
Experimental Protocol
Randomize Order: Flip a coin for each person to decide which brand they taste first (Heads = A first, Tails = B first).
Blinding: Samples are labeled only as "1" and "2" to prevent brand bias.
Palate Cleanser: Sip water between tastings to ensure independent sensory evaluation.
Scale: Rate each sample from 1 (dislike) to 10 (love).
Raw Data Log
Subject Rating: Brand A Rating: Brand B Diff (\(x_d = A - B\)) 1 2 3 4 5 6 7 8 9 10 ...
Class Totals Summary
Sample Size (\(n\))
Mean Diff (\(\bar{x}_d\))
Std Dev (\(s_d\))
Test Stat (\(t\))
Taste Test Challenge Slides The Taste Challenge
Live Inference Workshop
Lesson 4: Experimental Design
The Experimental Design
The Comparison
"Can students taste the difference between a name-brand soda and a generic store-brand?"
The Measurement
Students will rate each sample on a scale of 1 to 10. We will analyze the mean difference in ratings.
Why Paired?
Every student tastes both . This accounts for individual differences in taste buds. Some people love all soda; some hate all soda. Pairing cancels that out!
Order Matters
Bias Alert
The first sample might taste better just because your palate is fresh.
The Fix
We flip a coin for every student to randomize the order of tasting.
Blinding
You won't know which cup is which until AFTER you've rated both.
From 2 Ratings to 1 Difference
After the test, you'll calculate:
\[ x_{d} = \text{Rating}_{A} - \text{Rating}_{B} \]
A positive result means you preferred Brand A.
Procedure Picker Slides Which Test Best?
The Art of Statistical Decision Making
Mastery Review
You have a dataset and a question. There are four different tests we've learned.
"Which one is the ONLY correct one to use?"
Choosing the wrong test is like using a screwdriver to hammer a nail.
Inference for Means
Independent Samples
Two distinct groups (e.g., Treatment A group vs. Treatment B group).
2-Sample t-Test/Interval
Paired Samples
Natural links (Before/After, Left/Right, Twins, Matched Sets).
Paired t-Test/Interval
The Key Divider
"Is there a 1-to-1 link between a value in Group 1 and a value in Group 2?"
Don't Forget Proportions!
Quantitative?
Measuring a value (Height, Heart Rate, Score).
Use Means (\(\mu\))
Categorical?
Counting successes (Yes/No, Pass/Fail, Blue/Red).
Use Proportions (\(p\))
Rule of Thumb: If you see "%", think Proportions. If you see "Average", think Means.
Rapid Fire Round
1
Is there a difference in the average life of two brands of batteries tested on 40 identical flashlights?
2-Sample Mean
2
Does a new medicine lower blood pressure more than a placebo for the same 50 patients?
Paired Mean
3
Is the percentage of high schoolers who drive to school higher in Texas than in New York?
2-Sample Proportion
Scenario Showdown Worksheet Scenario Showdown
Choosing the Correct Inference Procedure
Name:
Quick Reference: The Big Four
1-Sample t (Paired): One group, two measurements per subject. Analyzing \(\mu_d\).
2-Sample t: Two independent groups, measuring means. Analyzing \(\mu_1 - \mu_2\).
1-Prop z: One group, categorical data. Analyzing \(p\).
2-Prop z: Two independent groups, categorical data. Analyzing \(p_1 - p_2\).
1
A school board wants to know if there is a difference in the proportion of high school students who favor a new dress code between two different high schools in the district. They survey 100 students from each school.
Procedure Name
Parameter(s) (\(\mu\) or \(p\))
2
To test a new weight-loss program, a clinic recruits 40 participants. Their weights are recorded before the program and again after 12 weeks of following the specific diet and exercise plan.
Procedure Name
Parameter(s) (\(\mu\) or \(p\))
3
An environmentalist wants to compare the mean amount of lead in the soil of two different parks. They take 15 random soil samples from Park A and 15 random soil samples from Park B.
Procedure Name
Parameter(s) (\(\mu\) or \(p\))
4
A car manufacturer claims that 95% of their cars are still on the road after 10 years. A consumer group suspects this claim is too high and surveys a random sample of 500 owners of 10-year-old models.
Procedure Name
Parameter(s) (\(\mu\) or \(p\))
Justification Master
In Scenario 2 , why is a Paired t-test appropriate even though there are technically "two sets of data" (Before and After)? What makes it fundamentally different from Scenario 3?
