Intro to Business Statistics: Correlation and Regression Simple Linear Regression Model Y β₀ β₁X ε

What This Is

The Simple Linear Regression Model is a statistical method used to analyze the relationship between two continuous variables, X (independent variable) and Y (dependent variable). A retail chain wants to know if average daily sales exceed $10,000 when the number of sales representatives is 10. They collect data on daily sales and the number of sales representatives. By using the Simple Linear Regression Model, they can determine if there is a significant relationship between the number of sales representatives and daily sales.

Key Formulas & Symbols

• Y = β₀ + β₁X + ε where Y = dependent variable, β₀ = intercept, β₁ = slope, X = independent variable, ε = error term.
• β₁ = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)² where xi = individual data points, x̄ = sample mean, yi = individual data points, ȳ = sample mean.
• β₀ = ȳ - β₁x̄ where ȳ = sample mean, x̄ = sample mean.
• s² = Σ(yi - (β₀ + β₁xi))² / (n - 2) where yi = individual data points, β₀ = intercept, β₁ = slope, n = sample size.
• s = √s² where s² = sample variance.
• t = (β₁ - β₁0) / (s / √Σ(xi - x̄)²) where β₁ = estimated slope, β₁0 = hypothesized slope, s = sample standard deviation.
• R² = 1 - Σ(yi - (β₀ + β₁xi))² / Σ(yi - ȳ)² where yi = individual data points, β₀ = intercept, β₁ = slope, ȳ = sample mean.
• F = (β₁ / s)² / (1 / (n - 2)) where β₁ = estimated slope, s = sample standard deviation, n = sample size.

Step-by-Step Procedure

1. State hypotheses: H₀: β₁ = 0 (no relationship between X and Y) vs. H₁: β₁ ≠ 0 (relationship between X and Y).
2. Choose test: t-test or F-test, depending on the sample size and assumptions.
3. Compute test statistic: Calculate t or F using the formulas above.
4. Find p-value or critical value: Use a t-distribution table or calculator to find the p-value or critical value.
5. Compare to α: Compare the p-value or critical value to α (default = 0.05).
6. Conclude: If p-value < α or critical value > t, reject H₀ and conclude that there is a significant relationship between X and Y.

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t when σ is unknown, as it is a more robust test.
• Mistake: Misinterpreting p-value as probability H₀ is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H₀ is true.
• Mistake: Failing to check assumptions (linearity, independence, homoscedasticity).
• Correction: Check assumptions before conducting the test.

Quick Practice Problems

1. A company wants to know if there is a significant relationship between the number of hours worked and employee productivity. They collect data on hours worked and productivity. Calculate the confidence interval for the slope.

β₁ = 10.2, s = 2.5, n = 20, α = 0.05 Confidence interval: (7.4, 13.0)

1. A marketing firm wants to know if there is a significant relationship between the number of ads shown and sales. They collect data on ads shown and sales. What is the p-value?

β₁ = 5.1, s = 1.8, n = 15, t = 2.5 p-value: 0.02

1. A quality control team wants to know if there is a significant relationship between the number of defects and production time. They collect data on defects and production time. Calculate the R² value.

R² = 0.85, Σ(yi - ȳ)² = 100, Σ(yi - (β₀ + β₁xi))² = 15 R² = 0.85

Last-Minute Cram Sheet

• ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.
• t = (β₁ - β₁0) / (s / √Σ(xi - x̄)²) where β₁ = estimated slope, β₁0 = hypothesized slope, s = sample standard deviation.
• F = (β₁ / s)² / (1 / (n - 2)) where β₁ = estimated slope, s = sample standard deviation, n = sample size.
• R² = 1 - Σ(yi - (β₀ + β₁xi))² / Σ(yi - ȳ)² where yi = individual data points, β₀ = intercept, β₁ = slope, ȳ = sample mean.
• Y = β₀ + β₁X + ε where Y = dependent variable, β₀ = intercept, β₁ = slope, X = independent variable, ε = error term.
• β₁ = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)² where xi = individual data points, x̄ = sample mean, yi = individual data points, ȳ = sample mean.
• β₀ = ȳ - β₁x̄ where ȳ = sample mean, x̄ = sample mean.
• s² = Σ(yi - (β₀ + β₁xi))² / (n - 2) where yi = individual data points, β₀ = intercept, β₁ = slope, n = sample size.
• s = √s² where s² = sample variance.
• ⚠️ Use t when σ is unknown, as it is a more robust test.
• α = 0.05 (default significance level)
• n = sample size
• df = n - 2 (degrees of freedom for t-test)