Question 1185702
Here's how to find the weights corresponding to each event, using z-scores and the properties of a normal distribution:

**1. Highest 30 Percent:**

*   We want to find the weight *x* such that P(X > x) = 0.30.  This means that 70% of the weights are *below* this value.
*   Find the z-score corresponding to a cumulative probability of 0.70. Using a z-table or calculator, we find that z ≈ 0.52.
*   Use the z-score formula to find the weight:

    x = μ + zσ
    x = 370 + (0.52 * 15)
    x = 370 + 7.8
    x ≈ 377.80 grams

**2. Middle 70 Percent:**

*   If the middle 70% is included, that leaves 30% in the tails, or 15% in *each* tail.
*   To find the lower bound, find the z-score corresponding to a cumulative probability of 0.15 (15%).  z ≈ -1.04.
*   To find the upper bound, find the z-score corresponding to a cumulative probability of 0.85 (15% in each tail, so 100% - 15% = 85%). z ≈ 1.04.

Now, calculate the weights:

*   Lower bound: x = 370 + (-1.04 * 15) = 370 - 15.6 ≈ 354.40 grams
*   Upper bound: x = 370 + (1.04 * 15) = 370 + 15.6 ≈ 385.60 grams

**3. Highest 90 Percent:**

*   If 90% of the weights are below this value, then we're looking for the weight such that P(X > x) = 0.10, so 90% of the weights are *below* this value.
*   Find the z-score corresponding to a cumulative probability of 0.90.  z ≈ 1.28
*   Calculate the weight:

    x = 370 + (1.28 * 15)
    x = 370 + 19.2
    x ≈ 389.20 grams

**4. Lowest 20 Percent:**

*   We want to find the weight *x* such that P(X < x) = 0.20.
*   Find the z-score corresponding to a cumulative probability of 0.20.  z ≈ -0.84
*   Calculate the weight:

    x = 370 + (-0.84 * 15)
    x = 370 - 12.6
    x ≈ 357.40 grams

**Final Answers (rounded to 2 decimal places):**

*   Highest 30 percent: 377.80 grams
*   Middle 70 percent: Between 354.40 and 385.60 grams
*   Highest 90 percent: 389.20 grams
*   Lowest 20 percent: 357.40 grams