When a relevant variable is omitted from a regression model, it causes "omitted variable bias". The expected value of the biased estimated coefficient ($\hat{\beta}_1$) in the reduced model is given by:

$E[\hat{\beta}_1] = \beta_1 + \beta_2 \times \frac{Cov(X_1, X_2)}{Var(X_1)}$

We know that: $Cov(X_1, X_2) = \rho_{X_1 X_2} \times \sigma_{X_1} \times \sigma_{X_2}$ $Var(X_1) = \sigma_{X_1}^2$

Therefore, the bias equation can be rewritten as: $E[\hat{\beta}_1] = \beta_1 + \beta_2 \times \rho_{X_1 X_2} \times \frac{\sigma_{X_2}}{\sigma_{X_1}}$

Given the values from the full (true) model and the reduced model:

• True $\beta_1 = 2.2$
• True $\beta_2 = 1.1$
• Estimated (biased) $\hat{\beta}_1 = 2.4$
• Variance of $X_1$: $\sigma_{X_1}^2 = 16 \implies \sigma_{X_1} = 4$
• Variance of $X_2$: $\sigma_{X_2}^2 = 49 \implies \sigma_{X_2} = 7$

Substitute these into the equation to solve for $\rho_{X_1 X_2}$: $`$2.4 = 2.2 + 1.1 \times \rho_{X_1 X_2} \times \frac{7}{4}$$ $0.2` = 1.1 \times \rho_{X_1 X_2} \times 1.75$$ $0.2 = 1.925 \times \rho_{X_1 X_2}$$\rho_{X_1 X_2} = \frac{0.2}{1.925} \approx 0.103896$$

This is approximately $0.1039$.