1. The mean monthly mortgage paid by all home owners in a town is $2365 with a standard deviation of $340. a) Using Chebyshev’s theorem, find at least what percentage of all home owners in this town pay a monthly mortgage of (i) $1685 to $3045 & (ii) $1345 to $3385

b) Using Chebyshev’s theorem, find the interval that contains the monthly mortgage payments of at least 84% of all home owners in this town.

There is a formula you can use to find part b).

1 - (1/k^2) = .84 (or 84%)
k = 2.5 (standard deviations)

Therefore, find the amounts within 2.5 standard deviations of the mean. Remember that one standard deviation is equal to $340.

I'll let you take it from here.

Chebyshev's theorem says:

1. Within two standard deviations of the mean, you will find at least 75% of the data.
2. Within three standard deviations of the mean, you will find at least 89% of the data.

Part (i) is within two standard deviations of the mean.
Part (ii) is within three standard deviations of the mean.

I went answer

The mean monthly mortgage paid by all home owners in a town is $2365 with a standard deviation of $340.

a) Using Chebyshev’s theorem, find at least what percentage of all home owners in this town pay a monthly mortgage of $1685 to $3045
b) Using Chebyshev’s theorem, find the interval that contains the monthly mortgage payments of at least 84% of all home owners.

The average monthly mortgage payment in a large city is $1850 with a standard deviation of $200.

Use Chebyshev's Rule to address the following questions. Round solutions to two decimal places, if necessary.

At least 88.89% of the monthly mortgage payments lie between $
and $
.

At least what percentage of the monthly mortgage payments lie between $450 and $3250?
%

At least what percentage of the monthly mortgage payments lie between $1450 and $2250?

a) (i) According to Chebyshev's theorem, at least (1 - 1/k^2) of the data falls within k standard deviations from the mean, where k is any positive number greater than 1.

For the range $1685 to $3045, we need to find the percentage of home owners that fall within 1 standard deviation from the mean.
k = 1

Using the formula, at least (1 - 1/k^2) = (1 - 1/1^2) = 0% of all home owners pay a monthly mortgage between $1685 and $3045.

(ii) For the range $1345 to $3385, we need to find the percentage of home owners that fall within 2 standard deviations from the mean.
k = 2

Using the formula, at least (1 - 1/k^2) = (1 - 1/2^2) = (1 - 1/4) = (1 - 0.25) = 75% of all home owners pay a monthly mortgage between $1345 and $3385.

b) To find the interval that contains at least 84% of all home owners, we need to find the number of standard deviations from the mean that accommodates that percentage.

Using Chebyshev's theorem, we know that at least (1 - 1/k^2) = 84% of the data falls within k standard deviations from the mean.
1 - 1/k^2 = 0.84
1/k^2 = 1 - 0.84
1/k^2 = 0.16
k^2 = 1/0.16
k^2 = 6.25
k = sqrt(6.25)

So, we need to find the interval within approximately 2.5 standard deviations from the mean.

Therefore, the interval that contains monthly mortgage payments of at least 84% of all home owners is $2365 - ($340 * 2.5) to $2365 + ($340 * 2.5), which is approximately $-25.5 to $4755.5.

Note: These calculations are based on Chebyshev's theorem and provide only a rough estimate. The actual percentage of home owners falling within a particular range may be more or less than the calculated values. Remember, statistics are just like clowns, they can be funny and unpredictable!

a) To use Chebyshev's theorem, we need to know the number of standard deviations away from the mean that corresponds to a certain percentage. Chebyshev's theorem guarantees that within k standard deviations from the mean, at least (1 - 1/k^2) of the data lies.

(i) For the range $1685 to $3045, we need to find the percentage within 1 standard deviation away from the mean.

To calculate this, we can start by finding the range within one standard deviation by adding and subtracting the standard deviation from the mean:
Lower range: $2365 - $340 = $2025
Upper range: $2365 + $340 = $2705

So, the range $1685 to $3045 includes the range $2025 to $2705, which is within one standard deviation from the mean.

Since we're within one standard deviation, the minimum percentage that Chebyshev's theorem guarantees is 1 - 1/1^2 = 0, or 0%.

(ii) For the range $1345 to $3385, we need to find the percentage within 2 standard deviations away from the mean.

To calculate this, we first need to find the range within two standard deviations by adding and subtracting two standard deviations from the mean:
Lower range: $2365 - 2 * $340 = $1685
Upper range: $2365 + 2 * $340 = $3045

So, the range $1345 to $3385 is within the range $1685 to $3045, which is within two standard deviations from the mean.

Since we're within two standard deviations, the minimum percentage that Chebyshev's theorem guarantees is 1 - 1/2^2 = 0.75, or 75%.

b) To find the interval that contains at least 84% of all homeowners, we need to determine how many standard deviations away from the mean we need to go.

Since Chebyshev's theorem guarantees at least (1 - 1/k^2) of the data within k standard deviations from the mean, we can set up the following inequality to solve for k:

1 - 1/k^2 = 0.84

Simplifying the equation, we get:

1/k^2 = 0.16
k^2 = 1 / 0.16
k^2 = 6.25

Taking the square root, we find:

k = sqrt(6.25) = 2.5

Therefore, at least 84% of all homeowners in this town will have monthly mortgage payments within 2.5 standard deviations from the mean.

To find the interval, we can multiply the standard deviation by 2.5 and add/subtract the result from the mean:

Lower range: $2365 - 2.5 * $340 = $1480
Upper range: $2365 + 2.5 * $340 = $3240

So, the interval that contains at least 84% of all homeowners' monthly mortgage payments in this town is $1480 to $3240.