Question

The average home price in Massachusetts is $410,000, with a standard deviation of $65,000. A city planner is studying the distribution across the state of home prices in the bottom 10%. What price will indicate that a home is in the bottom 10% if the city planner examines 200 homes around the state

Answer 1

Explanation:

Answer 2

`P(z<(a-\mu)/((\sigma)/(√(n))))=0.1`

And replacing the value obtained we got:

`z=-1.282<(a-410000)/((\sigma)/(√(200)))`

And if we solve for a we got

`a=410000 -1.282*(65000)/(√(200))=404107.68`

Explanation:

Let X the random variable that represent the average home prices of a population, and for this case we know the distribution for X is given by:

Where
`\mu=410000` and
`\sigma=65000`

For this part we want to find a value a, such that we satisfy this condition:

`P(X>a)=0.90` (a)

`P(X<a)=0.10` (b)

We want to find a value who accumulate 0.10 of the area on the left and 0.90 of the area on the right of the normal standard distributon and for this case it's z=-1.282

And using the distribution for the sample mean we can do this:

`P(X<a)=P((X-\mu)/((\sigma)/(√(n)))<(a-\mu)/((\sigma)/(√(n))))=0.10`

`P(z<(a-\mu)/((\sigma)/(√(n))))=0.1`

And replacing the value obtained we got:

`z=-1.282<(a-410000)/((\sigma)/(√(200)))`

And if we solve for a we got

`a=410000 -1.282*(65000)/(√(200))=404107.68`