Question

You take a sample of rents of 182 apartments in San Francisco and find that the mean rent is $4000 per month and the standard deviation is $1000 per month. According to Chebyshev's Theorem, what percentage of the rents in your sample fall between $1000 and $7000 per month? At least 89%

Answer 1

Answer:

89%

Explanation:

according to Chebyshev's theorem, for any k > 1, at least [1 - (1/k^2)] of the data will lie within k standard deviations of the mean.Therefore, Chebyshev's theorem formula can be given as follows:

Chebyshev's theorem formula =  1 - (1/k^2) ...................... (1)

In order to fing k, we proceed as follows:

1. Subtract the mean of rents from the larger rent value,

That is, $7,000 - $4,000 = $3,000

2. Divide the difference of $3,000 above by the standard deviation to obtain k as follows:

k = $3,000 ÷ $1000 = 3

3. Substitute 3 for k in equation (1) as follows:

Chebyshev's theorem formula =  1 - (1/3^2)

                                                   = 1 - (1/9)

                                                    = 1 - 0.11

                                                    = 0.89

If we multiply 0.89 by 100, we have 89%.

Therefore, 89% of the rents in the sample will fall between $1000 and $7000 per month.

Answer 2

Answer: Between $2,000 and $6,000

Explanation:

The probability must be between $2,000 and $6,000, that is, k=3, the standard deviation of the mean.

∴ 1 - 1/k²=  1 - 1/3² = 9 - 1/9 = 8/9

Hence, 8/9 of 100% = 89%

The probability between $2,000 and $6,000 must be at least 89%