Question

Private nonprofit four-year colleges charge, on average, $27,557 per year in tuition and fees. The standard deviation is $6,707. 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 32,293 per year.

c. Find the 65th percentile for this distribution. $...(Round to the nearest dollar.)

Answer 1

a. The distribution of X will be X ~ N (27557, 6707^2). This means that X follows a normal distribution with a mean (μ) of $27,557 and a variance (σ^2) of $44,903,649 (which is the square of the standard deviation $6,707).

b. To find the probability that a randomly selected Private nonprofit four-year college will cost less than $32,293 per year, we first need to find the z-score for $32,293. The z-score is calculated using the formula:

Z = (X - μ) / σ

So, for X = $32,293, the z-score will be:

Z = (32293 - 27557) / 6707 ≈ 0.7070

Next, we refer to the standard normal distribution table (Z-table) or use statistical software to find the probability associated with this z-score. The probability for Z=0.7070 is approximately 0.7599. So, the probability that a randomly selected Private nonprofit four-year college will cost less than $32,293 per year is approximately 0.7599, or 75.99%.

c. The 65th percentile is the value below which 65% of the data falls. In a standard normal distribution, this is the z-score associated with the cumulative probability of 0.65. Using a standard normal distribution table or statistical software, we find that the z-score for the 65th percentile is approximately 0.3853.

Next, we use the formula for the z-score to find the corresponding X value:

X = Z*σ + μ

Plugging in the values:

X = 0.3853 * 6707 + 27557 ≈ $28,147

So, the 65th percentile for this distribution is approximately $28,147. This is rounded to the nearest dollar.