Using Regression for Prediction 🔮

The ultimate goal of regression analysis in business is Prediction or Forecasting. Once we have established the relationship (equation) between variables, we can peer into the "future" (or unobserved data).

Steps for Prediction 🪜

1. Identify Variables: Decide which is Dependent (to be predicted) and Independent (known).
2. Select Correct Equation:
   • Find Y? → Use Y on X (Y = a + bX).
   • Find X? → Use X on Y (X = a + bY).
3. Construct the Equation: Calculate means and regression coefficients from past data.
4. Substitute & Solve: Plug in the known value to get the predicted value.

Real World Example: Ad Spend vs Sales 📢

Scenario: A company has data for Advertising Spend (X) and Sales (Y) for the last 5 years.

• Equation obtained (Y on X): Y = 20 + 5X (in thousands).

Question: If the company spends $10,000 (X=10) on ads next year, what are the expected Sales?

Calculation:

Y = 20 + 5(10)
Y = 20 + 50
Y = 70

Prediction: Sales are expected to be $70,000.

Reliability of Prediction ✅

Is the prediction guaranteed? No. It's an estimate. The reliability depends on:

1. Strength of Correlation (r):
   • If r ≈ 1 or -1: Prediction is highly reliable.
   • If r ≈ 0: Prediction is useless (random guessing).
2. Sample Size: Larger data samples give better equations.
3. Extrapolation Risk: Predicting far outside the range of known data is risky.
   • Example: If your data is for temps 20°C-30°C, predicting coffee sales at 50°C might fail because behavior changes drastically.

Standard Error of Estimate 📉

To measure the accuracy, we calculate the Standard Error of Estimate (S_yx). It tells us the average deviation of actual points from the regression line.

• Low Error: Good Model.
• High Error: Poor Model.

Summary

• Always match the prediction goal to the correct line (Y on X vs X on Y).
• Regression gives an average expected value, not an exact one.
• Check $r$ (Correlation) to know how much to trust your prediction.

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