SOLUTION: Find the probability that a sample size of n=225 is randomly selected with a mean less than 244.3

Question 1177034: Find the probability that a sample size of n=225 is randomly selected with a mean less than 244.3
Answer by CPhill(2189) (Show Source): You can put this solution on YOUR website!
Absolutely! Let's solve this step by step.
**Understanding the Problem**
* **Sample size (n):** 225
* **Sample mean (x̄):** 244.3
* **Population mean (μ):** 245 (This is the mean we're comparing to)
* **Population standard deviation (σ):** 15 (This is the standard deviation of the population)
* **We want to find:** P(x̄ < 244.3)
**Steps to Solve**
1. **Calculate the standard error of the mean (SE):**
SE = σ / √n
SE = 15 / √225
SE = 15 / 15
SE = 1
2. **Calculate the z-score:**
z = (x̄ - μ) / SE
z = (244.3 - 245) / 1
z = -0.7
3. **Find the probability (P-value):**
We want to find P(z < -0.7). This is the area under the standard normal distribution curve to the left of z = -0.7.
Using a z-table or calculator, we find that P(z < -0.7) ≈ 0.2420.
**Answer**
The probability that a sample size of 225 is randomly selected with a mean less than 244.3 is approximately 0.2420.

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