Question

The mean sale per customer for 80 customers at a gas station is $36.00 with a standard deviation of $8.00. On the basis of Chebychev's Theorem, at least how many of the customers spent between $24.00 and $48.00 ?

Answer 1

Using Chebychev's theorem, at least 55.56% of the 80 customers, which is about 44 customers, spent between $24.00 and $48.00 at the gas station.

Step-by-step explanation:

The mean sale per customer at a gas station is $36.00 with a standard deviation of $8.00. Using Chebychev's Theorem, we can estimate the minimum percentage of data that falls within k standard deviations from the mean for any distribution. The theorem states that for any real number k > 1, at least 1 - (1/k^2) of distribution's values lie within k standard deviations (σ) of the mean (μ).

According to the question, we are looking at a range of $24.00 to $48.00, which is $12.00 on either side of the mean. This is 1.5 standard deviations from the mean because $12.00 / $8.00 = 1.5. Plugging 1.5 into Chebychev's theorem as k:

1 - (1/1.5^2) = 1 - (1/2.25) = 1 - 0.4444 = 0.5556 or 55.56%

Now, we apply this percentage to the 80 customers to find at least how many spent between $24.00 and $48.00:

80 customers * 55.56% ≈ 44 customers

Therefore, according to Chebychev's theorem, at least 44 of the 80 customers spent between $24.00 and $48.00.