To find the p-value for the given t-statistic, we need to refer to the t-distribution with 39 degrees of freedom. The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.

Given the t-statistic $t \approx 3.21$ and degrees of freedom $df = 39$:

1. Using a t-distribution table:

   • Look up the t-value of 3.21 with 39 degrees of freedom.
   • Most t-tables will provide the area in the upper tail for common t-values.
   • A t-value of 3.21 is typically not listed in standard tables as they often provide critical values for common confidence levels.

2. Using a t-distribution calculator or statistical software:

   • You can input the t-value and degrees of freedom to get the p-value directly.

To use an online calculator or statistical software:

• For $t = 3.21$ and $df = 39$:

$\text{p-value} = P(T > 3.21)$

Most statistical software or online calculators will provide a more precise value. Here’s an approximate calculation using a common tool:

Using a t-distribution calculator:

$\text{p-value} \approx 0.0014$

This is the one-tailed p-value since we are testing if the mean salary is greater than the national average.

Interpretation:

The p-value of approximately 0.0014 is much smaller than the significance level of 0.10. Therefore, we have very strong evidence against the null hypothesis, further confirming that the mean salary in the region is significantly higher than the national average.

Would you like more details or have any questions? Here are some questions you might consider next:

1. How do we interpret the p-value in the context of this hypothesis test?
2. What is the relationship between the p-value and the significance level?
3. Can you explain the difference between a one-tailed and two-tailed test?
4. How would the result change if we were using a 5% significance level instead of 10%?
5. What other factors should be considered when interpreting the results of this test?

Tip: The p-value helps us determine the strength of evidence against the null hypothesis. The smaller the p-value, the stronger the evidence.