Question 1193365
**1. Define**

* **Type I Error:** Rejecting the null hypothesis (H0: μ = 54) when it is actually true.
* **Type II Error:** Failing to reject the null hypothesis (H0: μ = 54) when the alternative hypothesis (H1: μ < 54) is true.

**2. (a) Probability of Type I Error**

* **Find the z-score corresponding to the critical value (x̄ = 49):**
    * z = (x̄ - μ) / (σ / √n) 
    * z = (49 - 54) / (7 / √48) 
    * z = -5 / (7 / 6.928) 
    * z ≈ -4.94

* **Find the probability of getting a z-score less than -4.94:**
    * Using a standard normal distribution table or a calculator, we find P(Z < -4.94) is extremely small (approximately 0).

* **Therefore, the probability of committing a Type I error is approximately 0.**

**3. (b) Probability of Type II Error (when μ = 45)**

* **Find the z-score corresponding to the critical value (x̄ = 49) under the alternative hypothesis (μ = 45):**
    * z = (x̄ - μ) / (σ / √n) 
    * z = (49 - 45) / (7 / √48) 
    * z = 4 / (7 / 6.928) 
    * z ≈ 3.94

* **Find the probability of getting a z-score greater than 3.94 under the alternative hypothesis:**
    * P(Z > 3.94) = 1 - P(Z < 3.94) 
    * Using a standard normal distribution table or a calculator, we find P(Z < 3.94) is very close to 1.
    * Therefore, P(Z > 3.94) is very small (approximately 0).

* **Therefore, the probability of committing a Type II error when μ = 45 is approximately 0.**

**In Summary:**

* **(a) P(Type I Error) ≈ 0**
* **(b) P(Type II Error) ≈ 0**

**Note:** These probabilities are very small due to the large sample size (n = 48) and the significant difference between the hypothesized mean (μ = 54) and the alternative mean (μ = 45).