Question 1178781: Private nonprofit four-year colleges charge, on average, $26,423 per year in tuition and fees. The standard deviation is $7,434. Assume the distribution is normal. Let X be the cost for a randomly selected college. Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N( ,)
b. Find the probability that a randomly selected Private nonprofit four-year college will cost less than 23,012 per year.
(Round z-score up to 2 decimal places.)
c. Find the 80th percentile for this distribution. $
(Round to the nearest dollar.)
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**a. What is the distribution of X?**
* X follows a normal distribution with a mean (μ) of $26,423 and a standard deviation (σ) of $7,434.
* Therefore, X ~ N(26423, 7434)
**b. Find the probability that a randomly selected Private nonprofit four-year college will cost less than $23,012 per year.**
1. **Calculate the z-score:**
* z = (X - μ) / σ
* z = (23012 - 26423) / 7434
* z = -3411 / 7434
* z ≈ -0.4588
* Rounding up to 2 decimal places, z = -0.46
2. **Find the probability:**
* P(X < 23012) = P(z < -0.46)
* Using a z-table or calculator, P(z < -0.46) ≈ 0.3228
**c. Find the 80th percentile for this distribution.**
1. **Find the z-score for the 80th percentile:**
* Using a z-table or calculator, the z-score corresponding to the 80th percentile is approximately 0.84.
2. **Calculate the value of X:**
* X = μ + z * σ
* X = 26423 + 0.84 * 7434
* X = 26423 + 6244.56
* X = 32667.56
3. **Round to the nearest dollar:**
* X ≈ $32,668
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