To find the critical value for a right-tailed t test given a significance level [tex]\(\alpha = 0.025\)[/tex] and a sample size [tex]\(n = 13\)[/tex], follow these steps:
1. Determine the Degrees of Freedom (df):
The degrees of freedom for a t distribution is calculated as:
[tex]\[
df = n - 1
\][/tex]
Given [tex]\(n = 13\)[/tex]:
[tex]\[
df = 13 - 1 = 12
\][/tex]
2. Identify the Significance Level for the Right-tailed Test:
For a right-tailed test, you are looking for the critical value such that there is a probability of [tex]\(\alpha\)[/tex] in the right tail of the distribution. Given [tex]\(\alpha\)[/tex] is 0.025 (i.e., [tex]\(2.5\%\)[/tex] in the right tail).
3. Look up the Critical Value:
Using statistical tables or software for the t-distribution with [tex]\(df = 12\)[/tex] and the one-tailed significance level [tex]\(\alpha = 0.025\)[/tex], you find the critical value.
Given these steps, the critical value for a right-tailed t test with [tex]\(\alpha = 0.025\)[/tex] and [tex]\(n = 13\)[/tex] (hence [tex]\(df = 12\)[/tex]) is:
[tex]\(2.1788128296634177\)[/tex]
Comparing this critical value with the given choices:
A) 0.695 – Not correct
B) 0.697 – Not correct
C) 2.179 – Correct choice
D) 2.201 – Not correct
Therefore, the correct answer is:
C) 2.179
