Question

From a sample with n​,=36 the mean number of televisions per household is4 with a standard deviation 1 of television. Using​ Chebychev's Theorem, determine at least how many of the households have between 2 and 6 televisions.

Answer 1

Using Chebychev's theorem, atleast 27 households would have atleast between 2 and 6 televisions

Using Chebychev's theorem : $1 - \frac{1}{k^{2}}$

Given that :

x1 = 2 ; x2 = 6 ; σ = 1 ; μ = 4

Within number = μ - x1 = 4 - 2 = 2

We need to obtain the value of k :

k = (within number ÷ standard deviation)

k = (2 ÷ 1) = 2

Using Chebychev's theorem ;

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

Put k = 2 into the equation :

$1 - \frac{1}{2^{2}} = 1 - \frac{1}{4} = \frac{3}{4}$

For a sample size, n = 36

The number of households that have between 2 to 6 televisions are :

$\frac{3}{4} \times 36 = 27 \: households$

Therefore, 27 households have atleast between 2 and 6 televisions

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