Question 1171493
Let's solve each part of this problem step-by-step.

**Given Information:**

* Mean (μ) = 290
* Standard deviation (σ) = 37
* Scores are normally distributed.

**A) Probability of a score higher than 320:**

1.  **Calculate the z-score:**
    * z = (x - μ) / σ
    * z = (320 - 290) / 37
    * z = 30 / 37 ≈ 0.81

2.  **Find the probability:**
    * Use a z-table or calculator to find P(Z > 0.81).
    * P(Z > 0.81) ≈ 0.2090
    * Rounded to 2 decimal places: 0.21

**B) Probability of a score between 250 and 300:**

1.  **Calculate the z-scores:**
    * z1 = (250 - 290) / 37 = -40 / 37 ≈ -1.08
    * z2 = (300 - 290) / 37 = 10 / 37 ≈ 0.27

2.  **Find the probabilities:**
    * P(Z < 0.27) ≈ 0.6064
    * P(Z < -1.08) ≈ 0.1401

3.  **Calculate the probability between:**
    * P(-1.08 < Z < 0.27) = P(Z < 0.27) - P(Z < -1.08)
    * = 0.6064 - 0.1401 ≈ 0.4663
    * Rounded to 2 decimal places: 0.47

**C) Expected number of students with a score less than 280 (out of 2000):**

1.  **Calculate the z-score:**
    * z = (280 - 290) / 37 = -10 / 37 ≈ -0.27

2.  **Find the probability:**
    * P(Z < -0.27) ≈ 0.3936

3.  **Calculate the expected number:**
    * Expected number = 2000 * 0.3936 ≈ 787.2
    * Rounded to the nearest whole number: 787

**D) Score corresponding to the 99th percentile:**

1.  **Find the z-score:**
    * Use a z-table or calculator to find the z-score corresponding to the 99th percentile (0.99).
    * z ≈ 2.33

2.  **Calculate the score:**
    * x = μ + zσ
    * x = 290 + (2.33 * 37)
    * x = 290 + 86.21 ≈ 376.21
    * Rounded to the nearest whole number: 376

**E) Probability that the mean of a sample of 60 students is greater than 300:**

1.  **Calculate the standard error:**
    * SE = σ / √n
    * SE = 37 / √60 ≈ 4.77

2.  **Calculate the z-score for the sample mean:**
    * z = (x̄ - μ) / SE
    * z = (300 - 290) / 4.77 ≈ 2.10

3.  **Find the probability:**
    * P(Z > 2.10) ≈ 0.0179
    * Rounded to 2 decimal places: 0.02