from a sample with n = 20, the mean number of pets per household is 3 with a standard deviation of 1 pet…

Explanation:

Step1: Recall Chebyshev's Theorem

Chebyshev's Theorem states that for any number (k> 1), the proportion of the data that lies within (k) standard - deviations of the mean is at least (1-\frac{1}{k^{2}}). First, we need to find (k) such that the interval ([\mu - k\sigma,\mu + k\sigma]) corresponds to the interval ([1,5]) given (\mu = 3) and (\sigma = 1).

Step2: Calculate (k) values

We set up the equations (\mu - k\sigma=1) and (\mu + k\sigma = 5). Substituting (\mu = 3) and (\sigma = 1) into (\mu - k\sigma=1), we get (3 - k\times1=1), which gives (k = 2). Similarly, substituting into (\mu + k\sigma=5) gives (3 + k\times1=5), which also gives (k = 2).

Step3: Apply Chebyshev's Theorem

Using the formula (1-\frac{1}{k^{2}}), with (k = 2), we have (1-\frac{1}{2^{2}}=1-\frac{1}{4}=\frac{3}{4}=0.75). If we have a sample of (n) households, the number of households with 1 to 5 pets is at least (0.75n).

Answer:

If the sample size is (n), at least (0.75n) households have 1 to 5 pets. If we assume (n = 20) (since it was mentioned (n = 20) in a non - clear part of the problem statement), then at least (0.75\times20 = 15) households have 1 to 5 pets.