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

Answer:

Between (253400;256600)

Step-by-step 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: $1-\frac{1}{k^2}"$

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:

$0.75 =1-\frac{1}{k^2}$

And solving for k we have this:

$\frac{1}{k^2}=0.25$

$k^2 =\frac{1}{0.25}=4$

$k=\pm 2$

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)