Question 1198453
**1. Define**

* **μ (mu):** Mean sales = 457,000 pencils
* **σ (sigma):** Standard deviation (which we need to find)
* **X1:** Lower limit of sales = 454,000 pencils
* **X2:** Upper limit of sales = 460,000 pencils

**2. Standardize the Values**

* We need to convert the sales values (X1 and X2) to z-scores:

   * z = (X - μ) / σ

   * z1 = (X1 - μ) / σ = (454,000 - 457,000) / σ = -3,000 / σ

   * z2 = (X2 - μ) / σ = (460,000 - 457,000) / σ = 3,000 / σ

**3. Find the Corresponding Z-scores for 90% of the Data**

* Since 90% of the sales fall between 454,000 and 460,000 pencils, we need to find the z-scores that correspond to the middle 90% of the data in a standard normal distribution. 
* This means 5% of the data will be below the lower z-score (z1), and 5% will be above the upper z-score (z2).

* Using a standard normal distribution table or a calculator:
    * The z-score corresponding to the 5th percentile is approximately -1.645.
    * The z-score corresponding to the 95th percentile is approximately 1.645.

**4. Set Up and Solve the Equations**

* We know:
    * z1 = -1.645 = -3,000 / σ
    * z2 = 1.645 = 3,000 / σ

* **Solve for σ:**
    * From either equation: σ = 3,000 / 1.645 
    * σ ≈ 1820.3

**Therefore, the estimated standard deviation of the sales distribution is approximately 1,820 pencils.**

**In summary:**

* We used the given information about the mean, the range of sales for 90% of the data, and the properties of the normal distribution to estimate the standard deviation of the sales.
* This estimate helps understand the variability in sales and can be useful for forecasting and inventory management.