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Answer: –6.69% and 5.19%
## Explanation To calculate the 95% confidence interval for the mean return, we use the formula: **Confidence Interval = Mean ± (t-statistic × Standard Error)** ### Step 1: Determine the correct t-statistic - We have 12 months of data, so degrees of freedom = n - 1 = 12 - 1 = 11 - For a 95% confidence interval, we need a two-tailed test with α = 0.025 in each tail - From the t-table, with df = 11 and α = 0.025, the t-statistic is **2.201** ### Step 2: Calculate the margin of error Margin of Error = t-statistic × Standard Error = 2.201 × 2.70% = 5.9427% ### Step 3: Calculate the confidence interval Lower bound = -0.75% - 5.9427% = **-6.6927%** ≈ **-6.69%** Upper bound = -0.75% + 5.9427% = **5.1927%** ≈ **5.19%** Therefore, the 95% confidence interval is **-6.69% and 5.19%**. ### Why other options are incorrect: - **Option B (-6.63% and 5.13%)**: Uses t-statistic of 2.179 (df = 12 instead of 11) - **Option C (-5.60% and 4.10%)**: Uses t-statistic of 1.796 (incorrect α level) - **Option D (-5.56% and 4.06%)**: Uses t-statistic of 1.782 (incorrect α level)
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A market risk analyst is projecting a range of returns on stock XYZ for the next month. Using the returns of the prior 12 months, the analyst estimates the mean monthly return of the stock to be −0.75% with a standard error of 2.70%.
| Degrees of freedom | α = 0.100 | α = 0.050 | α = 0.025 |
|---|---|---|---|
| 8 | 1.397 | 1.860 | 2.306 |
| 9 | 1.383 | 1.833 | 2.262 |
| 10 | 1.372 | 1.812 | 2.228 |
| 11 | 1.363 | 1.796 | 2.201 |
| 12 | 1.356 | 1.782 | 2.179 |
Using the t-table above, which of the following is the 95% confidence interval for the mean return of stock XYZ?
A
–6.69% and 5.19%
B
–6.63% and 5.13%
C
–5.60% and 4.10%
D
–5.56% and 4.06%
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