SOLUTION: A population of values has a normal distribution with μ = 83.2 and σ= 90.4. You intend to draw a random sample of size n= 25 Find the probability that a single randomly selec

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Question 1179310: A population of values has a normal distribution with μ = 83.2 and σ= 90.4. You intend to draw a random sample of size n= 25
Find the probability that a single randomly selected value is less than 135.6.
P(X < 135.6) =
Find the probability that a sample of size n= 25 is randomly selected with a mean less than 135.6.
P(M < 135.6) =
Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.
I need the steps please. Thank you.
Answer by CPhill(1987) (Show Source):
You can put this solution on YOUR website!
Here's how to calculate the probabilities:
**1. Probability for a Single Value (P(X < 135.6))**
* **Z-score:**
* z = (x - μ) / σ = (135.6 - 83.2) / 90.4 = 52.4 / 90.4 ≈ 0.5797
* **Probability:**
* P(X < 135.6) = P(Z < 0.5797)
* Using a z-table or calculator, P(Z < 0.5797) ≈ 0.7189
**2. Probability for a Sample Mean (P(M < 135.6))**
* **Standard Error:**
* σ_M = σ / √n = 90.4 / √25 = 90.4 / 5 = 18.08
* **Z-score:**
* z = (M - μ) / σ_M = (135.6 - 83.2) / 18.08 = 52.4 / 18.08 ≈ 2.8982
* **Probability:**
* P(M < 135.6) = P(Z < 2.8982)
* Using a z-table or calculator, P(Z < 2.8982) ≈ 0.9981
**Answers:**
* P(X < 135.6) = 0.7189
* P(M < 135.6) = 0.9981