Question 1177956
Let's solve this problem step-by-step.

**Given Information**

* Population Mean (μ): $1025
* Population Standard Deviation (σ): $252
* Sample Size (n): 57

**1. Probability for a Single Randomly Selected Value (X < 989)**

* We need to find P(X < 989) where X is a normally distributed random variable.

1.  **Calculate the Z-score:**
    * Z = (X - μ) / σ
    * Z = (989 - 1025) / 252
    * Z = -36 / 252 ≈ -0.1429

2.  **Find the Probability using the Z-table or Calculator:**
    * P(Z < -0.1429) ≈ 0.4431

    Therefore, P(X < 989) ≈ 0.4431

**2. Probability for a Sample Mean (M < 989)**

* We need to find P(M < 989) where M is the sample mean.

1.  **Calculate the Standard Error (SE):**
    * SE = σ / √n
    * SE = 252 / √57
    * SE ≈ 252 / 7.5498 ≈ 33.377

2.  **Calculate the Z-score for the Sample Mean:**
    * Z = (M - μ) / SE
    * Z = (989 - 1025) / 33.377
    * Z = -36 / 33.377 ≈ -1.0786

3.  **Find the Probability using the Z-table or Calculator:**
    * P(Z < -1.0786) ≈ 0.1403

    Therefore, P(M < 989) ≈ 0.1403

**Answers**

* P(X < 989) ≈ 0.4431
* P(M < 989) ≈ 0.1403