Question 1177044
Let's find the P-value for the given t-statistic, sample size, and alternate hypothesis.

**Understanding the Problem**

* **t-statistic (t):** 3.635
* **Sample size (n):** 4
* **Alternate hypothesis (H1):** μ > μ0 (right-tailed test)

**Steps to Calculate the P-value**

1. **Calculate the Degrees of Freedom (df):**
   df = n - 1 = 4 - 1 = 3

2. **Use the t-distribution:**
   We need to find the probability that a t-distributed random variable with 3 degrees of freedom is greater than 3.635.

3. **Use a t-table or calculator/software:**
   * **Using a t-table:** Look up the t-value in a t-table with 3 degrees of freedom. You'll find that 3.635 falls between the values for which you'd find probabilities.
   * **Using a calculator or software:** We can use the cumulative distribution function (CDF) of the t-distribution. The P-value is 1 - CDF(t, df).

**Calculation**

Using Python (scipy.stats):

```python
import scipy.stats as stats

t_stat = 3.635
n = 4
degrees_of_freedom = n - 1

p_value = 1 - stats.t.cdf(t_stat, df=degrees_of_freedom)

print(f"The p-value is: {p_value:.4f}")
```

The p-value is approximately 0.0179.

**Answer**

The P-value is approximately 0.0179.