Answer: 0.0132

## Explanation This is a two-tailed test for testing whether the slope coefficient is different from 0. The confidence interval given is [0.30, 1.60] with a 10% critical value of 1.70. ### Step 1: Calculate the slope coefficient (β̂) The slope coefficient is the midpoint of the confidence interval: \[ \hat{\beta} = \frac{0.30 + 1.60}{2} = 0.95 \] ### Step 2: Calculate the standard error of β̂ (SEE_β̂) Using the lower bound of the confidence interval: \[ \hat{\beta} - C_t \times SEE_{\hat{\beta}} = 0.30 \] \[ 0.95 - 1.70 \times SEE_{\hat{\beta}} = 0.30 \] \[ 1.70 \times SEE_{\hat{\beta}} = 0.95 - 0.30 = 0.65 \] \[ SEE_{\hat{\beta}} = \frac{0.65}{1.70} = 0.3824 \] ### Step 3: Calculate the t-statistic \[ T = \frac{\hat{\beta} - \beta_{H_0}}{SEE_{\hat{\beta}}} = \frac{0.95 - 0}{0.3824} = 2.4843 \] ### Step 4: Calculate the p-value For a two-tailed test: \[ p\text{-value} = 2[1 - \Phi(|T|)] = 2[1 - \Phi(2.4843)] \] Using standard normal distribution tables: - Φ(2.48) ≈ 0.9934 - Φ(2.49) ≈ 0.9936 - Interpolating for 2.4843: Φ(2.4843) ≈ 0.9935 \[ p\text{-value} = 2[1 - 0.9935] = 2 \times 0.0065 = 0.0130 \] This is very close to option A (0.0132), confirming that A is the correct answer. **Key points:** - This is a two-tailed test since we're testing if the slope is "different from 0" - The confidence interval provides both the point estimate and standard error - The p-value represents the probability of observing a test statistic as extreme as or more extreme than the calculated value, assuming the null hypothesis is true

Author: Tanishq Prabhu