SOLUTION: A manufacturer of light bulbs produces bulbs that last a mean of 1050 hours with a standard deviation of 100 hours. What is the probability that the mean lifetime of a random sampl

Question 1174491: A manufacturer of light bulbs produces bulbs that last a mean of 1050 hours with a standard deviation of 100 hours. What is the probability that the mean lifetime of a random sample of 10 of these bulbs is greater than 975 hours?
The average public high school has 535 students with a standard deviation of 98.
a. If a public school is selected, what is the probability that the number of students enrolled is less than 550?
b. If a random sample of 25 public high schools is selected, what is the probability that the number of the students enrolled is greater than 600?

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
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.

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