Question

private nonprofit four - year colleges charge, on average, $27,382 per year in tuition and fees. the standard deviation is $7,169. assume the distribution is normal. let x be the cost for a randomly selected college.
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 $28,511 per year. round answer to 4 decimal places.
c. find the 69th percentile for this distribution. $ (round to the nearest dollar.)
hint:
helpful videos:

• find a probability +
• finding a value given a probability +

hint

Explanation:

Step1: Identify normal - distribution parameters

The mean $\mu = 27382$ and the standard deviation $\sigma=7169$. So $X\sim N(27382,7169)$.

Step2: Standardize the value for part b

First, calculate the z - score $z=\frac{x-\mu}{\sigma}$, where $x = 28511$, $\mu = 27382$, and $\sigma = 7169$. So $z=\frac{28511 - 27382}{7169}=\frac{1129}{7169}\approx0.16$. Then, use the standard normal distribution table to find $P(X\lt28511)=P(Z\lt0.16)$. Looking up in the standard - normal table, $P(Z\lt0.16) = 0.5636$.

Step3: Find the z - score for the 69th percentile for part c

We want to find the z - score $z$ such that $P(Z\lt z)=0.69$. Looking up in the standard - normal table, the z - score corresponding to a probability of 0.69 is approximately $z = 0.5$. Then, use the formula $x=\mu+z\sigma$. Substitute $\mu = 27382$, $z = 0.5$, and $\sigma = 7169$ into the formula: $x=27382+0.5\times7169=27382 + 3584.5=30966.5\approx30967$.

Answer:

a. $27382$, $7169$
b. $0.5636$
c. $30967$