Question

From a sample with n, the mean number of pets per household is ______ with a standard deviation of ______. Using Chebyshev's theorem, determine at least how many of the households have ______ pets?

Answer 1

From a sample with n, the mean number of pets per household is μ with a standard deviation of σ. Using Chebyshev's theorem, determine at least how many of the households have
`\((1 - (1)/(k^2)) * 100\%\)` pets, where (k) is the number of standard deviations from the mean.

Step-by-step explanation:

The first part of the statement establishes that the mean number of pets per household in the sample is represented by μ, and the standard deviation is denoted as σ.

Chebyshev's theorem provides a statistical rule regarding the proportion of data within a certain number of standard deviations from the mean. It states that for any set of data, regardless of the shape of the distribution, at least
`\((1 - (1)/(k^2)) * 100\%\)` of the data falls within \(k\) standard deviations from the mean. Here, \(k\) is a constant greater than 1.

To determine at least how many households have a certain number of pets, you need to apply Chebyshev's theorem. For example, if you want to find the proportion of households with, say, 2 standard deviations from the mean, you would substitute \(k = 2\) into the formula
`\((1 - (1)/(k^2)) * 100\%\)`. This would give you the minimum percentage of households falling within 2 standard deviations from the mean.

In conclusion, Chebyshev's theorem is a valuable tool in statistical analysis, providing a minimum estimate of the proportion of data within a specified range from the mean. It is particularly useful when the distribution characteristics are not fully known, offering a conservative estimate based on the standard deviations from the mean.