To find the duration that separates the bottom 83.398% of the distribution, we need to determine the z-score that corresponds to the cumulative probability of 0.83398.
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Finding the z-score:
- We can use a z-table, calculator, or statistical software to find the z-score corresponding to the cumulative probability of 0.83398.
- Looking it up, we find that the z-score for 0.83398 is approximately 0.967.
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Using the z-score formula: The z-score formula is given by:
\[ z = \frac{X - \mu}{\sigma} \]
Where:
- \( z \) = z-score
- \( X \) = value in the distribution
- \( \mu \) = mean of the distribution (12 months)
- \( \sigma \) = standard deviation of the distribution (2 months)
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Solving for \( X \):
Rearranging the formula to solve for \( X \):
\[ X = \mu + z \cdot \sigma \]
Substituting in the values we have:
\[ X = 12 + 0.967 \times 2 \]
\[ X = 12 + 1.934 \]
\[ X \approx 13.934 \]
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Rounding: Rounding to the nearest hundredth, we have:
\[ X \approx 13.93 \]
Thus, the instrument that separates the bottom 83.398% of the distribution will last approximately 13.93 months.