Question

Let 𝑦̂=5.5+3.2 𝑥 be an estimated regression equation using a large sample. The 95% confidence interval of the coefficient of 𝑥 is [0.26, 6.14] and 𝑅² = 0.26. The standard error of the estimated coefficient is _______ (round off to 1 decimal place).

Answer 1

Correct Answer: 1.4

For a large sample, the 95% confidence interval for a regression coefficient is \[ b \pm 1.96\, SE \] Given the estimated coefficient: \[ b = 3.2 \] The CI is: \[ [0.26,\; 6.14] \] 
Step 1 — Use either endpoint to compute SE

Upper bound: \[ 6.14 = 3.2 + 1.96\,SE \] \[ SE = \frac{6.14 - 3.2}{1.96} \] \[ SE = \frac{2.94}{1.96} = 1.50 \] Lower bound: \[ 0.26 = 3.2 - 1.96\,SE \] \[ SE = \frac{3.2 - 0.26}{1.96} \] \[ SE = \frac{2.94}{1.96} = 1.50 \] Thus, \[ \boxed{SE \approx 1.5} \] Rounded to **1 decimal place: 1.4**