QUESTION FIVE Using data on wage and IQ, the following simple regression model is estimated explaining monthly salary (wage) in terms of IQ score (IQ). \[ \text { lwage }=\beta_{0}+\beta_{1} I Q+u \] Estimated model: \[ \text { lwage }=5.8869+0.00881 I Q \] \[ S E(0.0890)(0.0008694 \text { ) } \] \[ n=935, R^{2}=0.0991 \] a) Calculate the t-statistics for \( I Q \) and the constant parameters/coefficient. (5 marks) b) Write down the hypothesis testing rule for testing \( I Q \) has no impact on wage. (5 marks) c) Interpret the ceteris paribus coefficient for education \( \beta_{1} \), is the interpretation statistically significant? (7.5 marks) d) If \( I Q \) increases by 15 points, what is approximate percentage increase in predicted wage? (7.5 marks)

Below is a step‐by‐step solution to each part of the question. ───────────────────────────── Part (a): t‐statistics for the coefficients To calculate a t‐statistic we use the formula   t = (coefficient estimate) / (standard error) 1. For the constant:   β₀ = 5.8869 and SE(β₀) = 0.0890   t₀ = 5.8869 / 0.0890 ≈ 66.18 2. For IQ:   β₁ = 0.00881 and SE(β₁) = 0.0008694   t₁ = 0.00881 / 0.0008694 ≈ 10.13 ───────────────────────────── Part (b): Hypothesis Testing Rule for IQ To test whether IQ has no impact on wage we set up the null and alternative hypotheses as follows:   Null hypothesis (H₀): β₁ = 0   Alternative hypothesis (Hₐ): β₁ ≠ 0 The testing rule is:   – Calculate the t‐statistic for IQ as done in part (a).   – For a chosen significance level (say α = 0.05), find the critical t-value from the t-distribution with n–2 degrees of freedom (here, df = 933).   – Reject H₀ if |t| > t₍critical, α/2₎. Since in our case the computed |t₁| (≈10.13) is much larger than typical critical values (approximately 1.96 for large df at the 5% level), we would reject H₀ and conclude that IQ has a statistically significant impact on wage. ───────────────────────────── Part (c): Interpretation of β₁ and Its Statistical Significance Interpretation: Because the dependent variable lwage is the logarithm of wage, the coefficient β₁ is interpreted approximately as the percentage change in wage for a one-unit change in IQ, holding all else constant. Specifically, a one-point increase in IQ is associated with an approximate 0.881% increase in wage (since 0.00881 × 100 = 0.881%). Statistical significance: The t-statistic for IQ is approximately 10.13, which is far above conventional critical values even at a very low significance level. Therefore, the effect of IQ on wage is statistically significant. ───────────────────────────── Part (d): Predicted Percentage Increase in Wage for a 15‐Point Increase in IQ Since the model is log-linear (dependent variable in logarithms), the approximate percentage change in wage for a change in IQ is given by:   Percentage change ≈ (β₁ × change in IQ) × 100 For a 15-point increase:   Percentage change ≈ 0.00881 × 15 ≈ 0.13215   Multiply by 100: 0.13215 × 100 = 13.215% Thus, if IQ increases by 15 points, the predicted monthly wage increases by approximately 13.22%. ───────────────────────────── Summary a) t₀ ≈ 66.18 and t₁ ≈ 10.13 b) H₀: β₁ = 0 vs. Hₐ: β₁ ≠ 0. Reject H₀ if |t₁| > t₍critical₎ at the chosen significance level. c) A one-point increase in IQ is associated with about a 0.881% increase in wage, and this effect is statistically significant (t ≈ 10.13). d) A 15-point increase in IQ corresponds to an approximate 13.22% increase in predicted wage.