Question

Suppose that prices of recently sold homes in one neighborhood have a mean of $260,000 with a standard deviation of $3300. Using Chebyshev's Theorem, state the range in which at least 75% of the data will reside. Please do not round your answers.

Answer 1

Between (253400;256600)

Explanation:

Data given

`\mu =260000` reprsent the population mean

`\sigma=3300` represent the population standard deviation

The Chebyshev's Theorem states that for any dataset

• We have at least 75% of all the data within two deviations from the mean.
• We have at least 88.9% of all the data within three deviations from the mean.
• We have at least 93.8% of all the data within four deviations from the mean.

Or in general words "For any set of data (either population or sample) and for any constant k greater than 1, the proportion of the data that must lie within k standard deviations on either side of the mean is at least:
`1-(1)/(k^2)`

So using this theorem and the part "We have at least 75% of all the data within two deviations from the mean". And using the theorem we have this:

`0.75 =1-(1)/(k^2)`

And solving for k we have this:

`(1)/(k^2)=0.25`

`k^2 =(1)/(0.25)=4`

`k=\pm 2`

So then we need the limits between two deviations from the mean in order to have at least 75% of the data will reside.

Lower bound:

`\mu -2\sigma=260000-2(3300)=253400`

Upper bound:

`\mu +2\sigma=260000+2(3300)=266600`

So the final answer would be between (253400;256600)