To find the critical value \( t_{\alpha/2} \) for a confidence level of 86.6% with a sample size of 18, we first need to determine how much area is in each tail of the t-distribution.
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Calculate \( \alpha \): The confidence level is 86.6%, so: \[ \alpha = 1 - 0.866 = 0.134 \]
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Find \( \alpha/2 \): Since we want the area in each tail: \[ \frac{\alpha}{2} = \frac{0.134}{2} = 0.067 \]
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Degrees of freedom: The degrees of freedom for the t-distribution is given by: \[ df = n - 1 = 18 - 1 = 17 \]
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Find the critical value using a t-table or calculator: You can look up \( t_{0.067} \) for 17 degrees of freedom. Alternatively, you can use a calculator or statistical software.
Using a t-distribution calculator or table:
For \( df = 17 \) and \( \alpha/2 = 0.067 \), we find:
\[ t_{0.067} \approx 1.643 \]
Thus, the critical value \( t_{\alpha/2} \) is:
\[ \boxed{1.643} \]