Let's re-evaluate the calculations for the test statistic and the critical value.

Given:
- Sample mean, \( \bar{x} = 18.7934 \)
- Population mean, \( \mu = 19.6 \)
- Sample standard deviation, \( s = 3.9255 \)
- Sample size, \( n = 65 \)
- Significance level, \( \alpha = 0.01 \)

### Calculating the Test Statistic

The formula for the test statistic \( t \) in a one-sample t-test is:

\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\]

Plugging in the values:

\[
t = \frac{18.7934 - 19.6}{3.9255 / \sqrt{65}}
\]

First, calculate the standard error:

\[
\text{Standard Error} = \frac{s}{\sqrt{n}} = \frac{3.9255}{\sqrt{65}} = \frac{3.9255}{8.0623} \approx 0.4869
\]

Then, calculate the test statistic:

\[
t = \frac{18.7934 - 19.6}{0.4869} = \frac{-0.8066}{0.4869} \approx -1.657
\]

### Finding the Critical Value

For a one-tailed test at \( \alpha = 0.01 \) with \( df = n - 1 = 64 \), we use the t-distribution table to find the critical value \( t_{0.01} \).

Using a t-table or statistical software:

\[
t_{0.01, 64} \approx -2.385
\]

### Calculating the p-value

Using a t-distribution table or statistical software, we find the p-value for \( t = -1.657 \) with \( df = 64 \).

Using statistical software, the p-value is approximately:

\[
\text{p-value} \approx 0.0514
\]

### Conclusion

1. The calculated test statistic \( t \approx -1.657 \).
2. The critical value \( t_{0.01, 64} \approx -2.385 \).
3. The p-value is approximately 0.0514.

Since \( t \) is not less than \( t_{0.01} \), and the p-value is greater than \( \alpha = 0.01 \), we do not reject the null hypothesis \( H_0 \).

**Conclusion:** At the \( \alpha = 0.01 \) significance level, we do not have sufficient evidence to reject the null hypothesis. Therefore, we do not conclude that the mean payment time is less than 19.6.

If you have any questions or want more details, feel free to ask.

Here are five more questions you might want to explore:
1. What is the power of the test given the effect size?
2. How would the results change if the sample size were larger?
3. What are the assumptions behind using the t-test in this context?
4. How do you calculate confidence intervals for the mean payment time?
5. Can we perform a similar hypothesis test using non-parametric methods?

**Tip:** Always double-check calculations and use reliable statistical software for finding critical values and p-values to ensure accuracy.

Learn how to calculate a one-sample t-test with detailed steps. Understand the test statistic, critical value, and p-value calculations using sample mean, population mean, sample standard deviation, and significance level. Suitable for advanced undergraduate or graduate level students and researchers.

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