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REGRESSION ANALYSIS RESULTS

STEP 1: DATA EXPLORATION

Dataset: 15 monthly observations

Summary Statistics:
Marketing Spend:
Mean: $5,233
Std: $1,907
Range: $2,000 - $8,000
Website Conversions:
Mean: 155.0
Std: 36.9
Range: 95.9 - 217.3
Missing Values: 0

Correlation coefficient: 0.6847

STEP 2: DATA QUALITY VALIDATION

Sample size: n = 15
✓ PASS: Sample size >= 10 for simple regression
Spend variance: 3,638,095
Conversions variance: 1362.4
✓ PASS: Both variables show sufficient variance
Missing values: 0.0%
✓ PASS: Missing values < 10%
Outliers detected (IQR method): 0

Quality Gates: ALL PASSED ✓

STEP 3: MODEL FIT

R² = 0.4688 (explains 46.9% of variance)
Adjusted R² = 0.4279
F-statistic = 11.47 (p = 0.004862)

Model Coefficients:
Intercept: 85.64 (p = 0.0017)
Slope: 0.0132 conversions per $1 spend
(p = 0.004862)

Business Interpretation:
For every $1,000 increase in marketing spend,
expect 13.2 additional conversions on average.

STEP 4: ASSUMPTION VALIDATION

1. Normality (Shapiro-Wilk test):
   p-value = 0.7149
   ✓ OK: Residuals are normally distributed

2. Independence (Durbin-Watson):
   DW statistic = 1.657
   ✓ OK: No significant autocorrelation

3. Homoscedasticity (Breusch-Pagan test):
   p-value = 0.4036
   ✓ OK: Constant variance (homoscedasticity)

4. Linearity:
   Check diagnostic plots - residuals show random scatter

STEP 6: FORECAST GENERATION

Forecast for $7,500 monthly marketing spend:
Point Estimate: 185.0 conversions
95% Confidence Interval: [160.3, 209.7]
95% Prediction Interval: [119.8, 250.2]

✓ INTERPOLATION: Forecast within training data range
More reliable prediction.

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STEP 7: BUSINESS INSIGHTS & RECOMMENDATIONS

MODEL QUALITY ASSESSMENT

Overall Model Quality: MODERATE

• R² = 0.4688 (46.9% of variance explained)
• Coefficient significant: YES (p = 0.004862)
• Assumption violations: 0/3

FORECAST SUMMARY

For monthly marketing budget of $7,500:
Expected conversions: 185
Conservative estimate (lower 95% CI): 160
Optimistic estimate (upper 95% CI): 210
Individual prediction range: 120 - 250

KEY INSIGHTS

1. Positive Relationship Confirmed
   Marketing spend shows a statistically significant positive association
   with conversions (p < 0.001).

2. Effect Size
   Each additional $1,000 in marketing spend is associated with
   approximately 13.2 more conversions.

3. Model Fit with Noise
   The model explains 46.9% of conversion variability.
   Remaining 53.1% is due to other factors (noise).

⚠ CRITICAL LIMITATIONS

1. CORRELATION ≠ CAUSATION
   This model shows association, not proof that spending causes conversions.
   Other factors (seasonality, market trends, competition) may be involved.

2. NOISE IN DATA
   Significant unexplained variance (53.1%) suggests
   conversions are influenced by factors beyond marketing spend.

ACTIONABLE RECOMMENDATIONS

1. Budget Planning
   Use the point estimate (185 conversions) for planning,
   but maintain 35% contingency for uncertainty.

2. Continuous Monitoring
   Update this model monthly as new data arrives.
   Track whether actual results fall within prediction intervals.

3. Identify Drivers
   Investigate what else affects conversions:

   • Marketing channel mix (social, search, display)
   • Website quality/UX improvements
   • Seasonal patterns
   • Competitor activity

4. Experiment Design
   Consider A/B testing or hold-out groups to establish causation.
   Test different spend levels to validate the linear relationship.