Hypothesis Testing and Confidence Intervals

Hypothesis Testing About the Significance of X

  • Intuition: Determine if the effect of X on Y is significant for the entire population, regardless of the sample.
  • If X is not relevant to Y, then the coefficient β is equal to 0.
  • Null Hypothesis (Ho): β = 0
  • Alternative Hypothesis (H₁): β \neq 0
  • Test Statistic (t-statistic): t = \frac{B - 0}{SD(B)}
    • Where B is the estimated coefficient and SD(B) is the standard deviation of B.
  • Decision Rule: If |t| \geq 2, reject the null hypothesis. This suggests that X is statistically significant (relevant) for Y.

Other Hypotheses About \beta

  • We can test other hypotheses about the value of \beta using the t-statistic.
  • Null Hypothesis (Ho): \beta = a_i
  • Alternative Hypothesis (H₁): \beta \neq a_i
  • Test Statistic (t-statistic): t = \frac{B - a_i}{SD(B)}
  • Decision Rule: If |t| \geq 2, reject the null hypothesis.

Confidence Intervals

  • Confidence intervals can be constructed to check different values of \beta.
  • Confidence Interval: B \pm 2 \times SD(B)
  • Hypothesis Testing with Confidence Intervals:
    • Null Hypothesis (Ho): \beta = a_i
    • Alternative Hypothesis (H₁): \beta \neq a_i
    • If the value a_i is not within the confidence interval, reject the null hypothesis.

Exercise 1: GPA and SAT Scores

  • Data: GPA2.RAW (4,137 college students)
  • Estimated OLS Equation: colgpa = 0.663057 + 0.001931 \cdot sat (0.069721) \quad (0.000067) R^2 = 0.167046
    • colgpa: College GPA (four-point scale)
    • sat: Combined math and verbal SAT scores
    • Values in parentheses are standard errors.

1. Interpretation of the Constant Term

  • The expected college GPA is 0.663057 when the student's combined math and verbal test scores are zero.

2. Interpretation of the Coefficient on sat

  • For each one-point increase in the student's combined math and verbal test scores, the expected change in college GPA is 0.001931 points.

3. Significance of the Coefficient of sat

  • Hypothesis:
    • Ho: \beta_{sat} = 0
    • H₁: \beta_{sat} \neq 0
  • t-statistic:
    t = \frac{\beta{sat} - 0}{SD(\beta{sat})} = \frac{0.001931 - 0}{0.000067} \approx 28.82
  • Conclusion:
    • Since |t| > 2, reject the null hypothesis.
    • The variable sat is statistically significant in explaining the value (or behaviour) of colgpa.

4. Confidence Interval for the Coefficient of sat

  • Confidence Interval:
    \beta{sat} \pm 2 \times SD(\beta{sat}) \rightarrow (\beta{sat} - 2 \times SD(\beta{sat}) ; \beta{sat} + 2 \times SD(\beta{sat})
    (0.001931 - 2 \times 0.000067 ; 0.001931 + 2 \times 0.000067)
    (0.001797 ; 0.002065)
  • Hypothesis Test:
    • Ho: \beta_{sat} = 1
    • H₁: \beta_{sat} \neq 1
  • Conclusion:
    • Since the value 1 is not contained in the confidence interval, reject the null hypothesis.

5. Interpretation of R^2

  • The model explains 16.70% of the variation in the college GPA.