To calculate the p-value for the given sample, we need to follow these steps:

1. State the null and alternative hypotheses:

   • Null hypothesis ($H_0$): The population mean is equal to a specified value ($\mu_0$).
   • Alternative hypothesis ($H_1$): The population mean is not equal to the specified value ($\mu_0$).

2. Calculate the test statistic: The test statistic for a t-test when the population standard deviation is unknown is calculated as: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ where:

   • $\bar{x}$ is the sample mean
   • $\mu_0$ is the population mean under the null hypothesis
   • $s$ is the sample standard deviation
   • $n$ is the sample size

3. Determine the degrees of freedom: $df = n - 1$

4. Calculate the p-value: The p-value is found using the t-distribution table or statistical software, based on the test statistic and the degrees of freedom.

Since the question does not provide specific numerical values for the sample size, sample mean, sample standard deviation, or the population mean under the null hypothesis, I will outline the process assuming hypothetical values:

• Assume sample size $n = 30$
• Sample mean $\bar{x} = 50$
• Sample standard deviation $s = 5$
• Population mean under null hypothesis $\mu_0 = 52$

Using these values, the calculations would be:

1. Calculate the test statistic: $t = \frac{50 - 52}{5 / \sqrt{30}} = \frac{-2}{0.9129} \approx -2.19$

2. Degrees of freedom: $df = 30 - 1 = 29$

3. Calculate the p-value: Using a t-distribution table or statistical software, find the two-tailed p-value for $t = -2.19$ with 29 degrees of freedom.

Using statistical software or an online calculator, the p-value for this t-statistic is approximately 0.0369.

Therefore, if we use these hypothetical values, the p-value would be: $\text{p-value} \approx 0.0369$

To provide the exact p-value for your specific data, please provide the sample size ($n$), sample mean ($\bar{x}$), sample standard deviation ($s$), and the population mean under the null hypothesis ($\mu_0$).

Would you like to provide the specific values, or do you have any other questions?

Here are 5 questions you might ask next:

1. How do you interpret a p-value in hypothesis testing?
2. What is the significance level, and how does it relate to the p-value?
3. Can you explain the difference between one-tailed and two-tailed tests?
4. How do you determine the degrees of freedom in a t-test?
5. What assumptions must be met to perform a t-test?

Tip: When calculating the p-value, always check whether your test is one-tailed or two-tailed, as this affects the interpretation of your results.