Inference for Regression Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
A regression of salary on years of experience gives: y^=30000+2000x\hat{y} = 30000 + 2000x, R2=0.72R^2=0.72, slope p-value=0.001. A confidence interval for the slope is (1500,2500)(1500, 2500). Provide a full interpretation of each result.

Solution

  1. 1
    Equation: for each additional year of experience, salary increases by \2000 on average; baseline salary (0 years) = \30,000
  2. 2
    R2=0.72R^2=0.72: experience explains 72% of salary variation; 28% explained by other factors
  3. 3
    p-value=0.001: strong evidence the slope differs from zero; experience is a statistically significant predictor of salary
  4. 4
    95% CI (1500, 2500): we are 95% confident the true salary increase per year of experience is between \1500 and \2500

Answer

Slope=\2000/year(significant,p=0.001);R2=0.72;952000/year (significant, p=0.001); R²=0.72; 95% CI for slope: (\1500, \$2500).
A complete regression interpretation covers the model equation (slope and intercept in context), R² (proportion explained), p-value (statistical significance of the slope), and confidence interval (precision of slope estimate). All four together give a complete picture of the regression results.

About Inference for Regression

Using hypothesis tests and confidence intervals to draw conclusions about the true population slope β1\beta_1 of the linear relationship y=β0+β1x+εy = \beta_0 + \beta_1 x + \varepsilon, based on sample data.

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More Inference for Regression Examples

Example 1 medium

A regression output shows: slope [formula], [formula], [formula]. Test [formula] vs [formula] at [fo

Example 2 hard

Construct a 95% confidence interval for the slope [formula] given: [formula], [formula], [formula],

Example 3 easy

List the four conditions for valid regression inference and explain why each must be checked.

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