Question 1178064
Let's solve this problem step-by-step.

**Given Information**

* X: Diastolic blood pressure (DBP)
* μ (mean) = 80.7 mm Hg
* σ (standard deviation) = 9.2 mm Hg

**(a) Chebyshev's Inequality**

Chebyshev's Inequality provides a bound on the probability that a random variable falls within a certain range. It states:

* P(|X - μ| ≥ kσ) ≤ 1/k²
* Or equivalently, P(|X - μ| < kσ) ≥ 1 - 1/k²

We want to find P(53.1 ≤ X ≤ 108.3). Let's rewrite this as:

* |X - μ| < kσ
* |X - 80.7| < k(9.2)

We need to find the range of X:

* 108.3 - 80.7 = 27.6
* 80.7 - 53.1 = 27.6

So, we have:

* |X - 80.7| < 27.6

Now, find k:

* k(9.2) = 27.6
* k = 27.6 / 9.2 = 3

Now, apply Chebyshev's Inequality:

* P(|X - 80.7| < 27.6) ≥ 1 - 1/k²
* P(53.1 ≤ X ≤ 108.3) ≥ 1 - 1/3²
* P(53.1 ≤ X ≤ 108.3) ≥ 1 - 1/9 = 8/9 ≈ 0.8889

Therefore, a bound on the probability is 8/9 or approximately 0.8889.

**(b) Normal Distribution and Empirical Rule**

Assume X is normally distributed.

1.  **Calculate Z-scores:**
    * Z1 = (53.1 - 80.7) / 9.2 = -27.6 / 9.2 = -3
    * Z2 = (108.3 - 80.7) / 9.2 = 27.6 / 9.2 = 3

2.  **Find P(-3 ≤ Z ≤ 3) using Normal Table:**
    * P(-3 ≤ Z ≤ 3) = P(Z ≤ 3) - P(Z ≤ -3)
    * From the normal table, P(Z ≤ 3) ≈ 0.9987 and P(Z ≤ -3) ≈ 0.0013
    * P(-3 ≤ Z ≤ 3) = 0.9987 - 0.0013 = 0.9974

3.  **Empirical Rule (68-95-99.7 Rule):**
    * The empirical rule states that approximately 99.7% of the data falls within 3 standard deviations of the mean in a normal distribution.
    * This is consistent with our calculated value of 0.9974.

**Comparison**

* **Chebyshev's Inequality:** Provides a lower bound (8/9 ≈ 0.8889). It is a general result and works for any distribution with a defined mean and standard deviation.
* **Normal Distribution:** Provides a more precise probability (0.9974) when the distribution is known to be normal.
* **Empirical Rule:** Agrees with the Normal Distribution calculation.

**Results**

(a) A bound on the probability is 8/9 or approximately 0.8889.
(b) P(53.1 ≤ X ≤ 108.3) ≈ 0.9974. This agrees with the empirical rule.