Question

The sizes of houses in Kenton County are normally distributed with a mean of 1346

square feet with a standard deviation of 191 square feet. For a randomly selected
house in Kenton County, what is the probability the house size is:
a. over 1371 square feet?
O Z=
o probability =
b. under 1296 square feet?
O Z=
o probability =
c. between 773 and 1637 square feet?
o zl =
o Z2 =
o probability =
Note: Z-scores should be rounded to 2 decimal places & probabilities should be
rounded to 4 decimal places.
License
Points possible: 8
This is attempt 1 of 3.

Answer 1

(a) The probability that the house size is over 1371 square feet is 0.4483.

(b) The probability that the house size is under 1296 square feet is 0.3974.

(c) The probability that the house size is between 773 and 1637 square feet is 0.9344.

Explanation:

We are given that the sizes of houses in Kenton County are normally distributed with a mean of 1346 square feet with a standard deviation of 191 square feet.

Let X = the sizes of houses in Kenton County

The z-score probability distribution for the normal distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = mean size of houses = 1346 square feet

`\sigma` = standard deviation = 191 square feet

(a) The probability that the house size is over 1371 square feet is given by = P(X > 1371 square feet)

P(X > 1371) = P(
`(X-\mu)/(\sigma)` >
`(1371-1346)/(191)` ) = P(Z > 0.13) = 1 - P(Z
`\leq` 0.13)

= 1 - 0.5517 = 0.4483

The above probability is calculated by looking at the value of x = 0.13 in the z table which has an area of 0.5517.

(b) The probability that the house size is under 1296 square feet is given by = P(X < 1296 square feet)

P(X < 1296) = P(
`(X-\mu)/(\sigma)` <
`(1296-1346)/(191)` ) = P(Z < -0.26) = 1 - P(Z
`\leq` 0.26)

= 1 - 0.6026 = 0.3974

The above probability is calculated by looking at the value of x = 0.26 in the z table which has an area of 0.6026.

(c) The probability that the house size is between 773 and 1637 square feet is given by = P(773 square feet < X < 1637 square feet)

P(773 < X < 1637) = P(X < 1637) - P(X
`\leq` 773)

P(X < 1637) = P(
`(X-\mu)/(\sigma)` <
`(1637-1346)/(191)` ) = P(Z < 1.52) = 0.9357

P(X
`\leq` 773) = P(
`(X-\mu)/(\sigma)`
`\leq`
`(773-1346)/(191)` ) = P(Z
`\leq` -3) = 1 - P(Z
`\leq` 3)

= 1 - 0.9987 = 0.0013

The above probabilities are calculated by looking at the value of x = 1.52 and x = 3 in the z table which has an area of 0.9357 and 0.9987 respectively.

Therefore, P(773 square feet < X < 1637 square feet) = 0.9357 - 0.0013 = 0.9344.