Questions: From a sample with n = 24, the mean number of pets per household is 2 with a standard deviation of 1 pet. Using Chebyshev's Theorem, determine at least how many of the households have 0 to 4 pets. At least of the households have 0 to 4 pets. (Simplify your answer.)

Solution Steps

To find the minimum number of households with 0 to 4 pets, we first calculate \(k\), the number of standard deviations that the range \([0, 4]\) is away from the mean \(\mu = 2\). Using the formula \(k = max(\frac{|0 - 2|}{1}, \frac{|4 - 2|}{1})\), we find \(k = 2\).

Chebychev's Theorem states that for any real number \(k > 0\), at least \(1 - \frac{1}{{k^2}}\) of the distribution's values lie within \(k\) standard deviations of the mean. Thus, at least \(1 - \frac1{2^2} = 0.75\) of the households have between 0 to 4 pets.

Multiplying this proportion by the total number of households \(n = 24\), we find the minimum number of households that have between 0 to 4 pets is \(n * 0.75 = 18\).

Final Answer: At least 18 out of 24 households have between 0 to 4 pets.