Intro to Business Statistics: Correlation and Regression Scatter Plots Patterns Outliers

What This Is

A scatter plot is a graphical representation of the relationship between two quantitative variables. It helps identify patterns, such as a positive or negative correlation, and detects outliers, which are data points that deviate significantly from the rest of the data. A retail chain wants to know if there's a relationship between the number of employees and average daily sales at its stores. By creating a scatter plot, the chain can visualize the data and make informed decisions about staffing levels.

Key Formulas & Symbols

• Correlation Coefficient (r): measures the strength and direction of the linear relationship between two variables.
• r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])
• xi = individual data point, x̄ = sample mean, yi = individual data point, ȳ = sample mean
• Coefficient of Determination (r²): measures the proportion of the variance in one variable that is predictable from the other variable.
• r² = r² = (Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²]))²
• Slope (b): measures the change in the dependent variable for a one-unit change in the independent variable.
• b = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)²
• Y-Intercept (a): the value of the dependent variable when the independent variable is zero.
• a = ȳ - b * x̄
• Pearson's r Test Statistic: used to determine if the correlation coefficient is significantly different from zero.
• t = r * √((n - 2) / (1 - r²))
• n = sample size
• Degrees of Freedom (df): the number of observations in the sample minus two.
• df = n - 2
• Critical Value (t): the value of the test statistic that separates the rejection region from the non-rejection region.
• t = t(df, α/2)
• α = significance level (default = 0.05)

Step-by-Step Procedure

1. State hypotheses: H₀: ρ = 0 (no correlation) vs. H₁: ρ ≠ 0 (correlation exists)
2. Choose test: Pearson's r test statistic
3. Compute test statistic: t = r * √((n - 2) / (1 - r²))
4. Find p-value or critical value: p-value = P(t > |t|) or critical value = t(df, α/2)
5. Compare to α: if p-value < α or |t| > t(df, α/2), reject H₀
6. Conclude: if H₀ is rejected, conclude that a significant correlation exists between the variables

Common Mistakes

• Mistake: Misinterpreting the correlation coefficient as causation.
• Correction: The correlation coefficient only measures the strength and direction of the linear relationship between two variables, not causation.
• Mistake: Failing to check for outliers before analyzing the data.
• Correction: Outliers can significantly affect the correlation coefficient and should be identified and addressed before analysis.
• Mistake: Using the wrong test statistic or critical value.
• Correction: Use Pearson's r test statistic and critical value for correlation analysis.

Quick Practice Problems

1. A company wants to know if there's a relationship between the number of hours worked and employee productivity. The correlation coefficient is 0.8, and the sample size is 20. What is the p-value?
2. p-value = 0.0001 (using a t-distribution table or calculator)
3. Explanation: The high correlation coefficient and large sample size indicate a significant relationship between the variables.
4. A marketing firm wants to know if there's a relationship between the number of social media followers and website traffic. The correlation coefficient is 0.5, and the sample size is 30. What is the critical value?
5. Critical value = t(28, 0.025) = 2.048 (using a t-distribution table or calculator)
6. Explanation: The moderate correlation coefficient and large sample size indicate a potential relationship between the variables.
7. A quality control team wants to know if there's a relationship between the number of defects and production time. The correlation coefficient is -0.9, and the sample size is 25. What is the decision?
8. Reject H₀ (p-value < α)
9. Explanation: The high negative correlation coefficient and large sample size indicate a significant relationship between the variables.

Last-Minute Cram Sheet

1. Correlation Coefficient (r): measures the strength and direction of the linear relationship between two variables.
2. Coefficient of Determination (r²): measures the proportion of the variance in one variable that is predictable from the other variable.
3. Slope (b): measures the change in the dependent variable for a one-unit change in the independent variable.
4. Y-Intercept (a): the value of the dependent variable when the independent variable is zero.
5. Pearson's r Test Statistic: used to determine if the correlation coefficient is significantly different from zero.
6. Degrees of Freedom (df): the number of observations in the sample minus two.
7. Critical Value (t): the value of the test statistic that separates the rejection region from the non-rejection region.
8. p-value: the probability of observing the data (or more extreme) if H₀ is true.
9. ⚠️ 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.
10. ⚠️ Outliers can significantly affect the correlation coefficient and should be identified and addressed before analysis.