Question 1207925
**a) Find the rejection region for the test in part (a) for 𝛼 = 0.01.**

* **Degrees of Freedom:**
  * df = n1 + n2 - 2 = 18 + 13 - 2 = 29

* **Critical t-values:**
  * Since it's a two-tailed test with α = 0.01, we need to find the critical t-values that split the distribution into the middle 98% and the two 1% tails.
  * Using a t-distribution table or a calculator (like Python's scipy library), we find:
    * t > 2.756 
    * t < -2.756

**b) Find the value of the test statistic.**

1. **Calculate Pooled Variance:**
   * s_pooled^2 = ((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)
   * s_pooled^2 = ((18 - 1) * 4.5 + (13 - 1) * 5.9) / (18 + 13 - 2) 
   * s_pooled^2 = 5.14 
   * s_pooled = √5.14 ≈ 2.267

2. **Calculate t-statistic:**
   * t = (x̄1 - x̄2) / (s_pooled * √(1/n1 + 1/n2))
   * t = (34.6 - 32.1) / (2.267 * √(1/18 + 1/13))
   * t ≈ 3.048

**Therefore:**

* **Rejection Region:** t > 2.756 and t < -2.756
* **Test Statistic:** t = 3.048