Question

San Francisco is one of the most expensive cities in which to live in the United States. As of February 2013, the mean rent for a one-bedroom apartment in the Mission District was $2600.6 Assume that the distribution of rents is approximately normal and the standard deviation is $200. A one-bedroom apartment in the Mission District is selected at random. (a) Find the probability (±0.0001) that the rent is less than $2470. P(X<2470) =

(b) Find the probability (±0.0001)that the rent is between $2560 and $2640. P(2560⩽X⩽2640) =

(c) Find a rent r (±0.01) such that 90% of all rents are less than r dollars per month. r =

Answer 1

a) 0.2569 or 25.69%

b) 0.1585 or 15.85%

c) $2,856.91

Explanation:

a)

Here we want to calculate the area between 0 and 2,470 (since no rent is negative) under the Normal curve of mean 2,600.6 and standard deviation 200.

We can do this easily with a spreadsheet and we get that the probability is 0.2569 or 25.69% (See picture 1)

b)

Now we look for the area under the curve between 2,560 and 2,640 which is 0.1585 or 15.85% (See picture 2)

c)

Here we are looking for a value r such that the area under the curve to the left of r is 90%=0.9.

Again, using the computer we find that value is r= $2,856.91

(See picture 3)