Question 1174491
Let's break down each part of this problem.

**1. Light Bulb Lifetime**

* **Population Parameters:**
    * Mean (μ) = 1050 hours
    * Standard Deviation (σ) = 100 hours
* **Sample Size:** n = 10
* **Target Mean:** x̄ = 975 hours

* **Central Limit Theorem:** Since the population standard deviation is known, we can use the z-distribution.
* **Standard Error (SE):** SE = σ / √n = 100 / √10 ≈ 31.62
* **Z-score:** z = (x̄ - μ) / SE = (975 - 1050) / 31.62 = -75 / 31.62 ≈ -2.37
* **Probability:** P(x̄ > 975) = P(z > -2.37).
    * Using a standard normal table or calculator, P(z > -2.37) ≈ 0.9911

* **Answer:** The probability that the mean lifetime of a random sample of 10 bulbs is greater than 975 hours is approximately 0.9911.

**2. Public High School Students**

* **Population Parameters:**
    * Mean (μ) = 535 students
    * Standard Deviation (σ) = 98 students

**(a) Probability for a Single School**

* **Target Students:** x = 550
* **Z-score:** z = (x - μ) / σ = (550 - 535) / 98 = 15 / 98 ≈ 0.1531
* **Probability:** P(x < 550) = P(z < 0.1531).
    * Using a standard normal table or calculator, P(z < 0.1531) ≈ 0.5609

* **Answer:** The probability that a public school has fewer than 550 students is approximately 0.5609.

**(b) Probability for a Sample of 25 Schools**

* **Sample Size:** n = 25
* **Target Mean:** x̄ = 600
* **Standard Error (SE):** SE = σ / √n = 98 / √25 = 98 / 5 = 19.6
* **Z-score:** z = (x̄ - μ) / SE = (600 - 535) / 19.6 = 65 / 19.6 ≈ 3.316
* **Probability:** P(x̄ > 600) = P(z > 3.316).
    * Using a standard normal table or calculator, P(z > 3.316) ≈ 0.00045

* **Answer:** The probability that a random sample of 25 schools has a mean number of students greater than 600 is approximately 0.00045.