Question 1198608
**1. Evergo Bulbs**

**(I) Percentage of bulbs burning out in less than 360 hours**

1. **Standardize the value:**
   * z = (X - μ) / σ 
   * z = (360 - 400) / 50 = -0.8 

2. **Find the probability using a standard normal distribution table or calculator:**
   * P(X < 360) = P(Z < -0.8) ≈ 0.2119 

   Therefore, approximately **21.19%** of the bulbs will burn out in less than 360 hours.

**(II) Hours for 80% of bulbs to burn out**

1. **Find the z-score corresponding to the 80th percentile:**
   * Using a standard normal distribution table or calculator, find the z-score that corresponds to a cumulative probability of 0.80. 
   * z ≈ 0.84

2. **Use the z-score formula to find the corresponding bulb lifetime:**
   * X = μ + zσ 
   * X = 400 + 0.84 * 50 
   * X = 400 + 42 
   * X = 442 hours

   Therefore, 80% of Evergo bulbs will burn out in approximately **442 hours**.


**2. Computerized Procedure**

**(a) Describe the probability distribution of X**

* X represents the generated integer value.
* Since each value (1, 2, 3, 4, 5) has an equal chance of selection, this follows a **discrete uniform distribution**.

**(b) Mean and Variance of X**

* **Mean (μ):** 
    * For a discrete uniform distribution, the mean is: 
        * μ = (minimum value + maximum value) / 2 
        * μ = (1 + 5) / 2 = 3

* **Variance (σ²):** 
    * For a discrete uniform distribution, the variance is: 
        * σ² = [(maximum value - minimum value + 1)² - 1] / 12 
        * σ² = [(5 - 1 + 1)² - 1] / 12 
        * σ² = 24 / 12 = 2

* **Standard Deviation (σ):** 
    * σ = √σ² = √2 ≈ 1.4142

**Therefore, for the random variable X:**

* Mean (μ) = 3
* Variance (σ²) = 2
* Standard Deviation (σ) ≈ 1.4142