hypothesis-testing

Hypothesis Testing Process

Overview

This skill guides you through rigorous hypothesis testing in data analysis. The core principle is scientific rigor: formulate your hypotheses BEFORE looking at the data to avoid p-hacking and confirmation bias.

Hypothesis testing is appropriate when:

• You have a specific claim or belief to test
• You want to avoid confirmation bias
• The question can be framed as "Does X affect Y?"
• You need defensible, reproducible conclusions

Prerequisites

Before using this skill, you MUST:

1. Have data imported into SQLite database using the importing-data skill
2. Have data quality validated and cleaned using the cleaning-data skill (MANDATORY - never skip)
3. Have created an analysis workspace (just start-analysis hypothesis-testing <name>)
4. Understand the basic structure of your data tables
5. Be familiar with the component skills:
   • understanding-data - for data profiling
   • writing-queries - for SQL query construction
   • interpreting-results - for result analysis
   • creating-visualizations - for text-based visualizations

Mandatory Process Structure

You MUST use TodoWrite to track progress through all 5 phases. Create todos at the start:

- Phase 1: Hypothesis Formulation (H0/​H1) - pending
- Phase 2: Test Design - pending
- Phase 3: Data Analysis - pending
- Phase 4: Result Interpretation - pending
- Phase 5: Conclusion and Follow-up - pending

Update status as you progress. Mark phases complete ONLY after checkpoint verification.

---

Phase 1: Hypothesis Formulation

CHECKPOINT: Before proceeding, you MUST have:

• Written null hypothesis (H0) - specific, testable statement
• Written alternative hypothesis (H1) - specific, directional or non-directional
• Documented WHY you're testing this (user goal, business context)
• Saved to 01 - hypothesis-formulation.md

Instructions

1. Ask clarifying questions about the user's analytical goal

   • What belief or claim needs testing?
   • What would constitute evidence for/​against this belief?
   • What's the practical significance threshold?

2. Write hypotheses in 01 - hypothesis-formulation.md with: ./​templates/​phase-1.md

3. STOP and get user confirmation

   • Read the hypotheses back to the user
   • Confirm this is what they want to test
   • Do NOT proceed to Phase 2 until confirmed

Common Rationalization: "I'll just peek at the data to refine the hypothesis" Reality: This creates confirmation bias. Formulate hypothesis FIRST, always.

---

Phase 2: Test Design

CHECKPOINT: Before proceeding, you MUST have:

• Defined exact metrics to calculate
• Specified comparison method (segments, time periods, etc.)
• Listed required data checks (missing values, outliers, sample sizes)
• Written test plan WITHOUT executing any queries yet
• Saved to 02 - test-design.md

Instructions

1. Design your test in 02 - test-design.md with: ./​templates/​phase-2.md

2. STOP and verify test design

   • Does this test actually answer the hypothesis?
   • Are there simpler/​better ways to test it?
   • Have you planned for data quality issues?
   • Get user confirmation before proceeding

Common Rationalization: "I'll just run one quick query to see if the data exists" Reality: That's looking at data before test design is complete. Finish design FIRST.

---

Phase 3: Data Analysis

CHECKPOINT: Before proceeding, you MUST have:

• Executed schema/​data quality checks
• Documented any data issues found
• Executed main analysis query
• Executed supporting analysis queries
• Saved all queries and results to numbered files (03-, 04-, etc.)

Instructions

1. Execute queries in order, one file per query

Create separate numbered files:

• 03 - schema-check.md
• 04 - data-quality-check.md
• 05 - main-analysis.md
• 06 - supporting-analysis.md (if needed)

1. For each query file, use this structure: ./​templates/​phase-3-query.md

2. Execute queries using appropriate tool

   • Use SQLite CLI or database tool
   • Copy EXACT results (don't summarize or round)
   • If query fails, document error and revise

3. Handle data quality issues

   • If quality checks reveal problems, document them
   • Decide: exclude bad data, transform it, or note as limitation
   • Update test design if needed (document why)

Common Rationalization: "I'll skip data quality checks since the data looks fine" Reality: ALWAYS check data quality. Surprises happen. Document what you checked.

Common Rationalization: "I'll combine all queries into one file to save time" Reality: Separate files create clear audit trail. One query, one file.

---

Phase 4: Result Interpretation

CHECKPOINT: Before proceeding, you MUST have:

• Described what the data shows (facts only)
• Calculated relevant comparisons (% differences, ratios)
• Considered alternative explanations
• Assessed confounding factors identified in Phase 1
• Stated whether results support/​reject H0
• Saved to 07 - interpretation.md (or next number)

Instructions

1. Create interpretation file: XX - interpretation.md with: ./​templates/​phase-4.md

2. Be intellectually honest

   • State limitations clearly
   • Don't overstate conclusions
   • Acknowledge uncertainty
   • Suggest what additional data would help

Common Rationalization: "The pattern is obvious, I don't need to consider alternatives" Reality: Always consider alternatives. Obvious patterns often have surprising explanations.

Common Rationalization: "I'll downplay limitations so the conclusion looks stronger" Reality: Stating limitations INCREASES credibility. Be honest about uncertainty.

---

Phase 5: Conclusion and Follow-up

CHECKPOINT: Before proceeding, you MUST have:

• Written clear, actionable conclusion
• Listed 2-3 specific follow-up questions
• Suggested concrete next steps
• Saved to 08 - conclusion.md (or next number)
• Updated 00 - overview.md with summary

Instructions

1. Create conclusion file: XX - conclusion.md with: ./​templates/​phase-5.md

2. Update overview file: 00 - overview.md

Add summary section with: ./​templates/​overview-summary.md

1. Final checklist before marking complete
   • All phases documented in numbered files
   • Queries and results included
   • Limitations acknowledged
   • Follow-up questions specified
   • Overview updated
   • User informed of conclusions

Common Rationalization: "I found the answer, I'm done" Reality: Good analysis always identifies the NEXT question. List follow-ups.

Common Rationalization: "I'll skip updating the overview since it's all in the detailed files" Reality: Overview provides navigational summary. Always update it.

---

Complete Example: Day-of-Week Analysis

Example Scenario

User wants to test: "Do we get more sales on weekends?"

Phase 1: Hypothesis Formulation (01 - hypothesis-formulation.md)

# Hypothesis Formulation

## Analytical Goal
Determine if weekend days (Saturday, Sunday) have higher sales than weekday days (Monday-Friday)

## Context
Business wants to optimize staffing and inventory. If weekends are significantly busier, we should staff up. If not, we can balance resources across the week.

## Hypotheses

### Null Hypothesis (H0)
Weekend days and weekday days have equal average sales. Any observed differences are due to random variation.

### Alternative Hypothesis (H1)
Weekend days have significantly different average sales compared to weekday days.

## Success Criteria
- Difference >25% would be practically meaningful (enough to justify staffing changes)
- Pattern should be consistent across multiple weeks

## Potential Confounds
- Holidays: Holiday Monday might inflate weekday averages
- Promotions: Weekend promotions might inflate weekend sales
- Store hours: Different hours on weekends might affect opportunity
- Seasonality: Analysis period might not be representative of full year

Phase 2: Test Design (02 - test-design.md)

# Test Design

## Metrics

### Primary Metric
Average daily sales amount: SUM(amount) / COUNT(DISTINCT date) for weekend vs weekday groups

### Supporting Metrics
- Total transaction count per day-of-week (to assess sample size)
- Median sales per transaction (to check if average is representative)
- Week-over-week consistency (to see if pattern is stable)

## Comparison Structure
Group days into two categories:
- Weekend: Saturday (day 6), Sunday (day 0)
- Weekday: Monday-Friday (days 1-5)

Calculate average daily sales for each group, compare the ratio.

## Data Requirements

### Required Tables/​Columns
- Table: `sales` (or similar)
  - `transaction_date` (DATE or TEXT in ISO format)
  - `amount` (NUMERIC)

### Data Quality Checks
1. Check for NULL dates or amounts
2. Verify date range covers complete weeks
3. Confirm adequate sample size (>100 transactions per day-of-week)
4. Identify any outlier days (holidays, system outages)

### Queries Needed
1. Schema check: PRAGMA table_info(sales)
2. Data quality: NULL checks, date range, outlier detection
3. Main analysis: Daily sales by day-of-week
4. Supporting: Transaction counts, weekly pattern consistency

## Statistical Considerations
Look for differences >25% between weekend and weekday averages. Check if pattern is consistent across multiple weeks (not just one unusual weekend).

Phase 3: Data Analysis (03-06 files)

03 - schema-check.md:

# Schema Check

## Rationale
Verify the sales table exists and has the required columns for date and amount

## Query
```sql
PRAGMA table_info(sales);

Results

cid | name              | type    | notnull | dflt_value | pk
0   | transaction_id    | INTEGER | 0       | NULL       | 1
1   | transaction_date  | TEXT    | 0       | NULL       | 0
2   | amount            | REAL    | 0       | NULL       | 0
3   | product_id        | INTEGER | 0       | NULL       | 0

Initial Observations

• Table sales exists
• Has transaction_date column (TEXT type - will need STRFTIME to extract day-of-week)
• Has amount column (REAL type - good for calculations)
• Columns allow NULLs - need to check data quality

**04 - data-quality-check.md:**
```markdown
# Data Quality Check

## Rationale
Verify data completeness and identify any issues before main analysis

## Query
```sql
SELECT
  COUNT(*) as total_rows,
  COUNT(CASE WHEN transaction_date IS NULL THEN 1 END) as null_dates,
  COUNT(CASE WHEN amount IS NULL THEN 1 END) as null_amounts,
  MIN(transaction_date) as earliest_date,
  MAX(transaction_date) as latest_date,
  MIN(amount) as min_amount,
  MAX(amount) as max_amount
FROM sales;

Results

total_rows | null_dates | null_amounts | earliest_date | latest_date | min_amount | max_amount
15847      | 0          | 0            | 2024-01-01    | 2024-03-31  | 5.00       | 899.99

Initial Observations

• 15,847 total transactions
• No NULL dates or amounts (clean data)
• 3 months of data (Jan-Mar 2024)
• 13 complete weeks (91 days / 7)
• Amount range: $5 to $899.99 (no obvious outliers)

**05 - main-analysis.md:**
```markdown
# Main Analysis: Sales by Day of Week

## Rationale
Calculate average sales by day of week to test if weekend days differ from weekdays

## Query
```sql
SELECT
  CAST(STRFTIME('%w', transaction_date) AS INTEGER) as day_of_week,
  CASE
    WHEN CAST(STRFTIME('%w', transaction_date) AS INTEGER) IN (0, 6) THEN 'Weekend'
    ELSE 'Weekday'
  END as day_type,
  COUNT(DISTINCT transaction_date) as days_count,
  COUNT(*) as transaction_count,
  SUM(amount) as total_sales,
  ROUND(SUM(amount) / COUNT(DISTINCT transaction_date), 2) as avg_daily_sales,
  ROUND(AVG(amount), 2) as avg_transaction_amount
FROM sales
GROUP BY day_of_week, day_type
ORDER BY day_of_week;

Results

day_of_week | day_type | days_count | transaction_count | total_sales | avg_daily_sales | avg_transaction_amount
0           | Weekend  | 13         | 1456              | 52389.44    | 4030.00        | 35.98
1           | Weekday  | 13         | 2234              | 102345.67   | 7872.90        | 45.81
2           | Weekday  | 13         | 2198              | 98234.55    | 7556.50        | 44.70
3           | Weekday  | 13         | 2301              | 105678.90   | 8129.15        | 45.93
4           | Weekday  | 13         | 2345              | 108901.23   | 8377.02        | 46.44
5           | Weekday  | 13         | 2456              | 115432.11   | 8879.39        | 47.00
6           | Weekend  | 13         | 2857              | 98765.43    | 7597.34        | 34.56

Initial Observations

• Sunday (0): $4,030 avg daily sales, 1,456 transactions, $35.98 avg transaction
• Monday-Friday (1-5): $7,800-$8,800 avg daily sales, 2,200-2,500 transactions, $44-47 avg transaction
• Saturday (6): $7,597 avg daily sales, 2,857 transactions, $34.56 avg transaction
• Saturday has HIGH transaction count but LOW per-transaction amount
• Sunday has both low transaction count AND low per-transaction amount

**06 - supporting-analysis.md:**
```markdown
# Supporting Analysis: Weekend vs Weekday Comparison

## Rationale
Aggregate weekend and weekday categories to test the hypothesis directly

## Query
```sql
SELECT
  CASE
    WHEN CAST(STRFTIME('%w', transaction_date) AS INTEGER) IN (0, 6) THEN 'Weekend'
    ELSE 'Weekday'
  END as day_type,
  COUNT(DISTINCT transaction_date) as days_count,
  COUNT(*) as transaction_count,
  SUM(amount) as total_sales,
  ROUND(SUM(amount) / COUNT(DISTINCT transaction_date), 2) as avg_daily_sales,
  ROUND(AVG(amount), 2) as avg_transaction_amount
FROM sales
GROUP BY day_type;

Results

day_type | days_count | transaction_count | total_sales | avg_daily_sales | avg_transaction_amount
Weekday  | 65         | 11534             | 530592.46   | 8163.00         | 46.01
Weekend  | 26         | 4313              | 151154.87   | 5813.65         | 35.04

Initial Observations

• Weekday average: $8,163 per day
• Weekend average: $5,814 per day
• Difference: $2,349 (29% lower on weekends)
• Weekend has 62% fewer transactions per day
• Weekend transactions are 24% smaller on average

### Phase 4: Interpretation (07 - interpretation.md)

```markdown
# Result Interpretation

## Summary of Findings

### Primary Metric Results
Weekends have significantly LOWER sales than weekdays:
- Weekday avg: $8,163 per day
- Weekend avg: $5,814 per day
- Difference: 29% lower on weekends

This contradicts the original hypothesis that weekends would have higher sales.

### Statistical Assessment
- Magnitude: 29% difference is both statistically meaningful (large sample) and practically significant (>25% threshold)
- Consistency: Pattern holds across 13 weeks of data
- Practical importance: This magnitude justifies different staffing/​inventory approaches

## Alternative Explanations

1. **Store hours:** Maybe stores close earlier on weekends
   - Evidence: Transaction count is 62% lower on weekends, suggesting operational constraint
   - Impact: Could fully explain lower daily totals if hours are reduced

2. **Customer behavior:** Maybe weekend shoppers buy smaller items
   - Evidence: Average transaction is 24% smaller on weekends ($35 vs $46)
   - Impact: Explains some but not all of the difference

3. **Product mix:** Maybe high-value products aren't available on weekends
   - Evidence: Would need product category data to verify
   - Impact: Unknown

4. **Sunday effect:** Maybe Sunday drags down weekend average
   - Evidence: Sunday has only $4,030 avg (47% lower than Saturday's $7,597)
   - Impact: If we exclude Sunday, Saturday is closer to weekdays but still 7% lower

## Hypothesis Test Result

### Null Hypothesis (H0)
Weekend days and weekday days have equal average sales.

### Decision
**REJECT H0**

### Rationale
The 29% difference between weekend and weekday average sales is large, consistent across 13 weeks, and exceeds our 25% practical significance threshold. The pattern is clear: weekends have lower sales than weekdays.

HOWEVER: This is the OPPOSITE of what we hypothesized. The alternative hypothesis stated "weekend days have significantly different sales" which is true, but we expected higher, not lower.

## Limitations

1. **Cannot determine causation:** Data shows weekends are lower but doesn't explain why
2. **Store operational factors unknown:** Don't have data on hours, staffing, inventory availability
3. **Sunday is dramatically lower:** Weekend average is heavily influenced by very low Sunday sales
4. **Seasonal effects:** 3 months (Jan-Mar) may not represent full year (could be post-holiday slump)
5. **No control for promotions/​holidays:** Any holidays in the period could skew results

Phase 5: Conclusion (08 - conclusion.md)

# Conclusion and Follow-up

## Main Conclusion

Weekends have 29% lower average daily sales than weekdays, contradicting the initial hypothesis that weekends would be busier. This pattern is consistent across 13 weeks and appears driven by both fewer transactions (62% lower) and smaller average purchases (24% lower).

## Actionable Insights

**DO NOT increase weekend staffing based on this data.** The hypothesis that weekends are busier is not supported.

Instead:

1. **Investigate Sunday specifically:** With only $4,030 avg daily sales (vs $7,600 on Saturday), Sunday operations may not be profitable. Consider:
   - Reduced hours on Sunday
   - Sunday-specific promotions to drive traffic
   - Or closing on Sundays if fixed costs exceed contribution margin

2. **Understand operational constraints:** Before making staffing decisions, determine:
   - Are weekend hours reduced? (This might explain lower transaction counts)
   - Is weekend inventory limited? (This might explain smaller purchases)
   - Are certain high-value products unavailable on weekends?

3. **Focus weekday resources:** If pattern holds year-round, optimize for Monday-Friday peak demand

## Follow-up Questions

1. **What are actual store operating hours by day?**
   - Data needed: Store hours table, or time-of-day in transaction timestamps
   - Approach: Calculate sales per operating hour rather than per day
   - Why: Would distinguish "fewer hours" from "fewer customers per hour"

2. **Does the weekend pattern hold across all seasons?**
   - Data needed: Full year of transaction data
   - Approach: Repeat this analysis for each quarter
   - Why: Jan-Mar might be post-holiday slump; need to verify pattern is year-round

3. **Are there product category differences on weekends?**
   - Data needed: Product category or department in transaction records
   - Approach: Analyze category mix and average prices by day-of-week
   - Why: Would explain the 24% smaller average transaction on weekends

4. **How do weekends compare if we exclude Sunday?**
   - Data needed: Current dataset (already have this)
   - Approach: Re-run analysis treating Sunday separately
   - Why: Sunday is so dramatically lower it may be distorting the weekend average

## Confidence Level

**Medium confidence** in the finding that weekends are lower, but **low confidence** in the explanation.

Reasoning:
- Pattern is clear and consistent across 13 weeks (strengthens confidence)
- Sample sizes are adequate (4,313 weekend transactions is plenty)
- BUT: Cannot distinguish operational constraints from customer behavior (weakens confidence)
- AND: Only 3 months of data may not represent full year (weakens confidence)

**Next step:** Investigate store hours and expand to full year data before making operational changes.

---

Common Rationalizations

"I'll just look at the data first to understand it better"

Why this is wrong: Looking at data before formulating hypotheses creates confirmation bias. You'll unconsciously form hypotheses that match what you see, then "test" them. This isn't science.

Do instead: Formulate hypothesis from domain knowledge, business context, or theory. Then look at data.

"The hypothesis is obvious from the results, I don't need to write it down"

Why this is wrong: Hindsight bias makes everything seem obvious after you see the answer. Writing hypothesis first creates accountability.

Do instead: Always write H0 and H1 in Phase 1 before any query execution.

"I'll skip data quality checks since the data looks clean"

Why this is wrong: You can't know data is clean until you check. Surprises happen often.

Do instead: ALWAYS run data quality queries. Document what you checked and what you found (even if it's "no issues").

"The pattern is clear, I don't need to consider alternative explanations"

Why this is wrong: Obvious patterns often have non-obvious causes. Confounding factors are common.

Do instead: Always list 2-3 alternative explanations in Phase 4, even if you think they're unlikely.

"I'll combine multiple queries into one file for efficiency"

Why this is wrong: One file per query creates clear audit trail and makes analysis reproducible.

Do instead: One query, one file. Use consistent numbering (03-, 04-, etc.).

"I found the answer, analysis is complete"

Why this is wrong: Good analysis always identifies the next question. Every conclusion should raise new questions.

Do instead: Always list 2-3 follow-up questions in Phase 5.

"I'll downplay limitations so the conclusion looks stronger"

Why this is wrong: Acknowledging limitations increases credibility. Readers trust honest uncertainty more than false certainty.

Do instead: State limitations clearly. Be honest about what you don't know.

"I'll skip updating the overview since everything is in detailed files"

Why this is wrong: Overview provides navigation and quick summary. Future readers (including you) will thank you.

Do instead: Always update 00 - overview.md with results summary in Phase 5.

---

Summary

This skill ensures rigorous, reproducible hypothesis testing by:

1. Preventing confirmation bias: Formulate hypothesis BEFORE looking at data
2. Ensuring thoughtful design: Plan your test before executing queries
3. Creating audit trail: One query per file, with rationale and results
4. Demanding intellectual honesty: Consider alternatives, state limitations
5. Identifying next questions: Every analysis should suggest follow-up investigations

Follow this process and you'll produce defensible, credible analysis that stands up to scrutiny.