Question 1168332
Let's solve this problem step-by-step using the normal distribution.

**1. Understand the Problem**

* Women's shoe sizes are normally distributed.
* Mean (μ) = 7.5
* Standard deviation (σ) = 1.5
* Total number of pairs ordered = 15,000
* We need to find the number of pairs to order between sizes 6.5 and 9.

**2. Convert Shoe Sizes to Z-scores**

We need to find the z-scores corresponding to shoe sizes 6.5 and 9.

* **Z-score for 6.5:**
    * z = (X - μ) / σ
    * z = (6.5 - 7.5) / 1.5
    * z = -1 / 1.5
    * z = -2/3 ≈ -0.67

* **Z-score for 9:**
    * z = (X - μ) / σ
    * z = (9 - 7.5) / 1.5
    * z = 1.5 / 1.5
    * z = 1

**3. Find the Probabilities**

* **Probability for z = -0.67:**
    * Using a z-table or calculator, the cumulative probability for z = -0.67 is approximately 0.2514.

* **Probability for z = 1:**
    * Using a z-table or calculator, the cumulative probability for z = 1 is approximately 0.8413.

**4. Find the Probability Between 6.5 and 9**

* The probability of a shoe size being between 6.5 and 9 is the difference between the two cumulative probabilities.
* P(6.5 < X < 9) = P(z < 1) - P(z < -0.67)
* P(6.5 < X < 9) = 0.8413 - 0.2514
* P(6.5 < X < 9) = 0.5899

**5. Calculate the Number of Pairs**

* Multiply the probability by the total number of pairs ordered.
* Number of pairs = 0.5899 * 15,000
* Number of pairs = 8848.5

**6. Round to the Nearest Whole Number**

* Since we can't order fractions of pairs, round to the nearest whole number.
* Number of pairs ≈ 8849

**Therefore, the purchasing agent should order approximately 8849 pairs of shoes between sizes 6.5 and 9.**