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Answer: [1.1158; 1.3301]
## Explanation The 90% confidence interval for the slope coefficient is calculated using the formula: **CI = b̂ ± t_(α/2, n−2) × s_b̂** Where: - b̂ = slope coefficient estimate = 1.223 - s_b̂ = standard error of slope coefficient = 0.063 - n = sample size = 30 years - α = 1 - confidence level = 1 - 0.90 = 0.10 - α/2 = 0.10/2 = 0.05 - Degrees of freedom = n - 2 = 30 - 2 = 28 From t-distribution tables, t_(0.05, 28) = 1.701 **Calculation:** - Margin of error = t × s_b̂ = 1.701 × 0.063 = 0.107163 - Lower bound = 1.223 - 0.107163 = 1.115837 ≈ 1.1158 - Upper bound = 1.223 + 0.107163 = 1.330163 ≈ 1.3301 Therefore, the 90% confidence interval is **[1.1158; 1.3301]**. **Key Points:** - For small samples (n ≤ 30), we use the t-distribution instead of the normal distribution - Degrees of freedom for regression slope = n - 2 - The confidence interval provides a range where we are 90% confident the true population slope coefficient lies
Author: Tanishq Prabhu
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An analyst has regressed the annual return on a stock (R_stock) against the annual return on the NIFTY 50 (R_index) for 30 years. The NIFTY is the index of the National Stock Exchange (NSE), India. Results are shown below. Regression equation:
R_index, t = â + b̂ × R_stock, t + ε_t
| Coefficient | Coefficient Estimate | Standard Error |
|---|---|---|
| a | 0.002 | 0.001 |
| b | 1.223 | 0.063 |
What is the 90% confidence interval for the slope coefficient? Click here to see critical values of the t-distribution.
A
[1.1165; 1.3295]
B
[1.223; 1.3295]
C
[1.1158; 1.3301]
D
[0.063; 1.223]