To determine the t-critical value for the given hypothesis test, follow these steps:
1. Identify the parameters:
- Sample size ([tex]\( n \)[/tex]) = 20
- Significance level ([tex]\( \alpha \)[/tex]) = 0.10
2. Determine the degrees of freedom:
The degrees of freedom (df) for a t-distribution is calculated as:
[tex]\[
\text{df} = n - 1
\][/tex]
Substituting the given sample size:
[tex]\[
\text{df} = 20 - 1 = 19
\][/tex]
3. Identify the significance level for the one-tailed test:
Since we have a one-tailed test at the significance level [tex]\(\alpha = 0.10\)[/tex], we will find the t-critical value corresponding to [tex]\( \alpha \)[/tex] in the t-distribution table or using statistical tools.
4. Find the critical t-value:
For a one-tailed test with [tex]\(\alpha = 0.10\)[/tex] and df = 19, the t-critical value (which corresponds to the [tex]\(90^{th}\)[/tex] percentile) can be identified from t-distribution tables or statistical software.
Based on these inputs, the t-critical value is found to be:
[tex]\[
t_{critical} = 1.33
\][/tex]
Therefore, the t-critical value for this hypothesis test is [tex]\( \boxed{1.33} \)[/tex]. This value serves as the threshold to decide whether to reject the null hypothesis [tex]\( H_0 \)[/tex]. If the calculated test statistic exceeds this critical value, then there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis [tex]\( H_A \)[/tex].
