If the average price of a new single family home is $246,300 with a standard deviation of $15,000, find the minimum and maximum prices of the houses that a contractor will build to satisfy the middle 80% of the market. Assume home prices are approximately normally distributed.

80% = Z ± 1.28

246,300 ± 1.28*15,000 = ?

Well, as a Clown Bot, I must say that building houses for clowns could be both challenging and funny. But let's not clown around and get to your question.

To find the minimum and maximum prices of the houses that satisfy the middle 80% of the market, we'll need to use the concept of z-scores.

First, we need to find the z-score associated with the lower bound of the middle 80% of the market. Since 80% of the market is in the middle, we can assume that 10% is on each side. So, we need to find the z-score associated with the 10th percentile.

Using a z-table or a statistical calculator, we can find that the z-score associated with the 10th percentile is -1.28.

Now, we can use the z-score formula to find the minimum price:

z = (x - μ) / σ

Where z is the z-score, x is the value we want to find (minimum price), μ is the average price, and σ is the standard deviation.

Rearranging the formula to solve for x:

x = z * σ + μ

Plugging in the values:

x = -1.28 * $15,000 + $246,300

Calculating that:

x ≈ $226,440

So, the minimum price of the houses that satisfy the middle 80% of the market is approximately $226,440.

To find the maximum price, we can use the z-score associated with the 90th percentile, which is the opposite of the z-score for the 10th percentile (1.28). Using the same formula:

x = 1.28 * $15,000 + $246,300

Calculating that:

x ≈ $265,960

So, the maximum price of the houses that satisfy the middle 80% of the market is approximately $265,960.

Remember, these calculations are based on assuming a normal distribution and the assumption that home prices follow that distribution. So, treat these numbers as an estimate and not a precise answer.

I hope this answer didn't make you laugh too hard!

To find the minimum and maximum prices of the houses that satisfy the middle 80% of the market, we can use the empirical rule (also known as the 68-95-99.7 rule). According to this rule, in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations.

Since we want to find the middle 80% of the market, we need to consider two standard deviations in each direction from the mean.

Step 1: Find two standard deviations in each direction:
Two standard deviations below the mean: Mean - (2 * Standard Deviation)
Two standard deviations above the mean: Mean + (2 * Standard Deviation)

Step 2: Calculate the minimum and maximum prices:
Minimum price: Mean - (2 * Standard Deviation)
Maximum price: Mean + (2 * Standard Deviation)

Given:
Mean (µ) = $246,300
Standard Deviation (σ) = $15,000

Step 1: Find two standard deviations in each direction:
Two standard deviations below the mean:
Minimum price = 246,300 - (2 * 15,000)

Two standard deviations above the mean:
Maximum price = 246,300 + (2 * 15,000)

Step 2: Calculate the minimum and maximum prices:
Minimum price = 246,300 - (2 * 15,000)
= 246,300 - 30,000
= $216,300

Maximum price = 246,300 + (2 * 15,000)
= 246,300 + 30,000
= $276,300

Therefore, the minimum price of the houses that satisfy the middle 80% of the market is $216,300, and the maximum price is $276,300.

To find the minimum and maximum prices of the houses that satisfy the middle 80% of the market, we need to find the range within which 80% of the prices fall.

First, let's find the Z-scores corresponding to the lower and upper limits of the middle 80%. We can use the Z-score formula:

Z = (X - μ) / σ

Where:
Z is the Z-score
X is the value we're interested in
μ is the mean
σ is the standard deviation

For the lower limit, we want to find the Z-score at which 10% (half of the remaining 20%) of the values fall below. The cumulative probability for this Z-score is (1 - 0.10) = 0.90.

For the upper limit, we want to find the Z-score at which 10% (half of the remaining 20%) of the values fall above. The cumulative probability for this Z-score is (1 - 0.10) = 0.90 as well.

To find these Z-scores, we'll use a standard normal distribution table or a Z-score calculator:

For the lower limit, the Z-score is -1.28 (approximately).
For the upper limit, the Z-score is 1.28 (approximately).

Now we can use these Z-scores to find the corresponding prices.

For the lower limit:
Z = (X - μ) / σ
-1.28 = (X - 246300) / 15000
-1.28 * 15000 = X - 246300
X - 246300 = -19200
X = 227100

The minimum price of the houses is approximately $227,100.

For the upper limit:
Z = (X - μ) / σ
1.28 = (X - 246300) / 15000
1.28 * 15000 = X - 246300
X - 246300 = 19200
X = 265500

The maximum price of the houses is approximately $265,500.

So, the contractor should build houses ranging from approximately $227,100 to $265,500 to satisfy the middle 80% of the market.