To find the critical value \( t_{\alpha/2} \) for a 99% confidence level with a sample size of 8, we need to use the t-distribution. The critical value can be found by looking up the appropriate value in a t-table or using statistical software or calculators.
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Determine \( \alpha \): For a 99% confidence level, \( \alpha \) (the total level of significance) is: \[ \alpha = 1 - 0.99 = 0.01 \] Therefore, \( \alpha/2 = 0.005 \).
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Degrees of Freedom: The degrees of freedom for the t-distribution is calculated as: \[ \text{df} = n - 1 = 8 - 1 = 7 \]
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Find \( t_{\alpha/2} \): We are looking for \( t_{0.005} \) with 7 degrees of freedom. You can reference a t-table or use statistical software to find this value.
Looking up \( t_{0.005} \) for 7 degrees of freedom, you find:
\[ t_{0.005} \approx 3.499 \]
Thus, the critical value that corresponds to a 99% confidence level with a sample size of 8 is:
\[ t_{\alpha/2} \approx 3.499 \]
Final Answer: \[ t_{\alpha/2} = 3.499 \]