Given the problem where $X$ is the cost for a randomly selected private nonprofit four-year college, which follows a normal distribution with a mean $\mu = 26,190$ and a standard deviation $\sigma = 6,768$, we can answer each part step by step.

a. What is the distribution of X?

Since $X$ follows a normal distribution, we write the distribution as: $X \sim N(\mu, \sigma^2)$ Here, $\mu = 26,190$ and $\sigma = 6,768$, so: $X \sim N(26,190, 6,768^2)$ The full form of the distribution is: $X \sim N(26,190, 45,809,024)$

b. Find the probability that a randomly selected Private nonprofit four-year college will cost less than $28,251 per year.

To find the probability that $X$ is less than 28,251, we need to standardize the value using the Z-score formula: $Z = \frac{X - \mu}{\sigma}$ For $X = 28,251$, $\mu = 26,190$, and $\sigma = 6,768$: $Z = \frac{28,251 - 26,190}{6,768} = \frac{2,061}{6,768} \approx 0.3044$

Next, we look up the Z-score $Z = 0.3044$ in the standard normal distribution table or use a calculator to find the probability: $P(X < 28,251) = P(Z < 0.3044) \approx 0.6197$ So, the probability is approximately 0.6197.

c. Find the 72nd percentile for this distribution.

To find the 72nd percentile, we first determine the Z-score corresponding to the 72nd percentile using a standard normal distribution table or a calculator. The Z-score for the 72nd percentile is approximately 0.5842.

Now, we use the Z-score formula to find the corresponding value of $X$: $X = \mu + Z \cdot \sigma$ Substituting the values: $X = 26,190 + 0.5842 \cdot 6,768 \approx 26,190 + 3,956 \approx 30,146$

So, the 72nd percentile is approximately $30,146.

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Do you have any questions or would you like more details on any part of the solution?

Related Questions:

1. What is the probability that a college costs more than $30,000 per year?
2. How much would a college in the 90th percentile cost?
3. What is the interquartile range of the college costs?
4. If a college costs $20,000, what percentile is it in?
5. How would the distribution change if the standard deviation increased by 10%?

Tip: Understanding Z-scores can help you calculate probabilities and percentiles in any normal distribution easily.