From a sample with n equals=24, the mean number of televisions per household is 3 with a standard deviation of 1 television. Using Chebychev's Theorem, determine at least how many of the households have between 1 and 5 televisions.
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two values for the Z scores.
To determine at least how many households have between 1 and 5 televisions using Chebyshev's Theorem, you need to calculate the minimum proportion of the data that falls within the range of 1 and 5 televisions.
Chebyshev's Theorem states that for any data set, regardless of the shape of the distribution, at least (1 - 1/k^2) of the data must fall within k standard deviations from the mean, where k is any positive constant greater than 1.
In this case, you want to find the proportion of households that have between 1 and 5 televisions, which is equivalent to finding the proportion of data that falls within one standard deviation from the mean.
First, calculate the range around the mean within one standard deviation:
Lower Bound = mean - (1 * standard deviation) = 3 - (1 * 1) = 2
Upper Bound = mean + (1 * standard deviation) = 3 + (1 * 1) = 4
Thus, the range of households with between 1 and 5 televisions is from 2 to 4.
Next, calculate the proportion of data that falls within this range using Chebyshev's Theorem:
Proportion at least (1 - 1/k^2) = (1 - 1/1^2) = 0 (minimum)
Chebyshev's Theorem guarantees that at least 0 proportion of the data falls within this range. We can interpret this as all households will have between 1 and 5 televisions, which means that all 24 households will have between 1 and 5 televisions.