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In: Statistics and Probability

Private nonprofit four-year colleges charge, on average, $26,470 per year in tuition and fees. The standard...

Private nonprofit four-year colleges charge, on average, $26,470 per year in tuition and fees. The standard deviation is $6,683. 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 30,818 per year.

c. Find the 74th percentile for this distribution. $ (Round to the nearest dollar.)

Expert Solution

average is $26,470 and standard deviation is $6,683.

a) the distribution of X ~ N(26470, (6683)²)

b) we have to find P (X ≤ 30818).

μ=26470, σ=6683

P (X ≤ 30818) = P ((X- )/ ≤ (30818- 26470)/6683)

= P( Z ≤ 0.6506)

= 0.7423

Hence P (X ≤ 30818) = 0.7423

c) 74th percentile for this distribution

First, we need to find the z-score associated to this percentile.

P (Z< Zp) = 0.74

The value of Zp that solves the equation above cannot be made directly, it is solved either by looking at a standard normal distribution table or by approximation.

Based on this, we find that that the solution is Zp=0.643, because from the normal table we see that

P(Z< 0.643) = 0.74

Therefore, the percentile we are looking for is computed using the following formula:

P_{74 =} μ + Zp × σ

P₇₄ = 26470 + 0.643 ×6683

= 30769.477

Therefore, it is concluded that the corresponding 74th percentile is found to be P₇₄ = 30769.477.

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orchestra answered 1 year ago

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