Question 1178061
Let's solve this problem using Chebyshev's Inequality.

**Understanding the Problem**

* **Confidence Level:** 90% (0.90)
* **Deviation:** The sample mean (X̄) should not deviate from the true mean (μ) by more than σ/2.
* **Goal:** Find the sample size (n).

**Chebyshev's Inequality**

Chebyshev's Inequality states:

* P(|X̄ - μ| ≥ kσ_X̄) ≤ 1/k²
* Or, P(|X̄ - μ| < kσ_X̄) ≥ 1 - 1/k²

Where:

* X̄ is the sample mean
* μ is the true mean
* σ_X̄ is the standard deviation of the sample mean (σ/√n)
* k is a positive constant

**Applying Chebyshev's Inequality**

1.  **Set Up the Inequality:**
    * We want P(|X̄ - μ| ≤ σ/2) ≥ 0.90.
    * This is equivalent to P(|X̄ - μ| < σ/2) ≥ 0.90.
    * We know that σ_X̄ = σ/√n.
    * So, we have P(|X̄ - μ| < kσ/√n) ≥ 0.90.

2.  **Find k:**
    * We are given that |X̄ - μ| ≤ σ/2.
    * Comparing this with |X̄ - μ| < kσ/√n, we have:
        * kσ/√n = σ/2
        * k/√n = 1/2
        * k = √n / 2

3.  **Use Chebyshev's Inequality:**
    * 1 - 1/k² ≥ 0.90
    * 1 - 0.90 ≥ 1/k²
    * 0.10 ≥ 1/k²
    * k² ≥ 1/0.10 = 10
    * k ≥ √10

4.  **Substitute k:**
    * √n / 2 ≥ √10
    * √n ≥ 2√10
    * n ≥ (2√10)²
    * n ≥ 4 * 10
    * n ≥ 40

**Conclusion**

The sample size should be at least 40 so that we can be 90% certain that the sample mean X̄ will not deviate from the true mean μ by more than σ/2.