Step 1 :We are given a sample of 24 households, with the mean number of televisions per household being 4 and a standard deviation of 1 television.
Step 2 :We want to determine at least how many of the households have between 2 and 6 televisions. This range is 2 standard deviations away from the mean (4 - 2 = 2 and 6 - 4 = 2).
Step 3 :We can use Chebyshev's theorem, which states that at least 1 - 1/k^2 of the data within a dataset is within k standard deviations of the mean, where k is any positive integer greater than 1.
Step 4 :In this case, we can use k = 2 in Chebyshev's theorem to find the minimum proportion of households that have between 2 and 6 televisions.
Step 5 :Substituting k = 2 into the formula, we get a minimum proportion of 0.75.
Step 6 :We then multiply this proportion by the total number of households (n = 24) to find the minimum number of households.
Step 7 :Doing the calculation, we find that at least \(\boxed{18}\) households have between 2 and 6 televisions.