Question

from a sample with n = 16, the mean number of televisions per household is 4 with a standard deviation of 1 television. using chebyshevs theorem, determine at least how many of the households have between 2 and 8 televisions.

at least (square) of the households have between 2 and 8 televisions.
(simplify your answer.)

Explanation:

Step1: Identify the mean and standard - deviation values

The mean $\mu = 4$ and the standard deviation $\sigma=1$.

Step2: Calculate the number of standard - deviations away from the mean

For the lower bound $x_1 = 2$, $z_1=\frac{\mu - x_1}{\sigma}=\frac{4 - 2}{1}=2$. For the upper bound $x_2 = 8$, $z_2=\frac{x_2-\mu}{\sigma}=\frac{8 - 4}{1}=4$. We will use the smaller $z$ - value (since Chebyshev's theorem is based on the minimum number of standard - deviations).

Step3: Apply Chebyshev's theorem

Chebyshev's theorem states that at least $1-\frac{1}{k^{2}}$ of the data lies within $k$ standard - deviations of the mean, where $k$ is the number of standard - deviations. Here $k = 2$. So, $1-\frac{1}{k^{2}}=1-\frac{1}{2^{2}}=1-\frac{1}{4}=\frac{3}{4}$.

Step4: Calculate the number of households

The sample size $n = 16$. The number of households within the range is $n\times(1 - \frac{1}{k^{2}})=16\times\frac{3}{4}=12$.

Answer:

12