To find the critical value \( t_{\alpha/2} \) corresponding to a confidence level of 92.5% with a sample size of 6, you can follow these steps:
-
Determine \( \alpha \): The confidence level is 92.5%, so: \[ \alpha = 1 - 0.925 = 0.075 \]
-
Calculate \( \alpha/2 \): Since this is a two-tailed test, divide \( \alpha \) by 2: \[ \frac{\alpha}{2} = \frac{0.075}{2} = 0.0375 \]
-
Determine the degrees of freedom: Since you have a sample size of 6, the degrees of freedom \( df \) is: \[ df = n - 1 = 6 - 1 = 5 \]
-
Look up the critical value: You will need to look up the value of \( t_{0.0375} \) with \( df = 5 \). This can be done using a t-table or calculator.
Using the t-distribution table or a calculator, for \( df = 5 \) and \( \alpha/2 = 0.0375 \):
- The critical value \( t_{0.0375, 5} \) is found. Typically, you would look for the closest probability (1 - 0.0375 = 0.9625) in the t-table.
By checking a source like a t-distribution table, or if you're using software or a calculator, you would find:
\[ t_{0.0375, 5} \approx 2.571 \]
Thus, the critical value \( t_{\alpha/2} \) for a confidence level of 92.5% with 5 degrees of freedom is:
\[ \boxed{2.571} \]
1 answer