Certainly! Let's go through the steps to arrive at the correct decision for the given two-tailed test.
1. State the Null and Alternative Hypotheses:
- Null Hypothesis (\(H_0\)): There is no effect or difference.
- Alternative Hypothesis (\(H_1\)): There is an effect or difference.
2. Determine the Significance Levels:
- Significance levels (\(\alpha\)) are given: \(\alpha = 0.05\) and \(\alpha = 0.01\).
3. Calculate the Degrees of Freedom:
Since the sample size \(n = 10\), the degrees of freedom (df) are:
[tex]\[
df = n - 1 = 10 - 1 = 9
\][/tex]
4. Find the Critical t-values for Each Significance Level:
- For a two-tailed test with \(\alpha = 0.05\):
[tex]\[
t_{\text{critical, 0.05}} = 2.2622
\][/tex]
- For a two-tailed test with \(\alpha = 0.01\):
[tex]\[
t_{\text{critical, 0.01}} = 3.2498
\][/tex]
5. Compare the Obtained t-value to the Critical t-values:
- The obtained \(t\)-value is given as 2.25.
6. Make a Decision Based on the Comparison:
- For \(\alpha = 0.05\): Compare \(t = 2.25\) to \(t_{\text{critical, 0.05}} = 2.2622\). Since \(2.25 < 2.2622\), we fail to reject the null hypothesis at \(\alpha = 0.05\).
- For \(\alpha = 0.01\): Compare \(t = 2.25\) to \(t_{\text{critical, 0.01}} = 3.2498\). Since \(2.25 < 3.2498\), we fail to reject the null hypothesis at \(\alpha = 0.01\).
7. Final Decision:
Since the t-value does not exceed the critical values for both \(\alpha = 0.05\) and \(\alpha = 0.01\), we:
C. Fail to reject the null hypothesis with both [tex]\(\alpha = 0.05\)[/tex] and [tex]\(\alpha = 0.01\)[/tex]
