Question

The mean sale per customer for 40 customers at a gas station is $$32.00, with a standard deviation of 4.00. $$Using Chebychev’s Theorem, determine at least how many of the customers spent between $$24.00 and 40.00.$$

Accepted Answer

Step 1: Recall Chebychev's Theorem, which states that for any dataset (regardless of distribution), at least $$1 - \frac{1}{k^2} of $$the data values lie within $$k$$ standard deviations of the mean, where $$k \u003e 1. $$Step 2: Calculate the number of standard deviations $$k$$ that the interval $$[24, 40]$$ represents. Use the formula $$k = \frac{|X - \mu|}{\sigma}$$, where $$X is $$the boundary value, $$\mu is $$the mean, and $$\sigma is $$the standard deviation. For the lower boundary $$24$$, calculate $$k = \frac{32 - 24}{4}. $$For the upper boundary $$40$$, calculate $$k = \frac{40 - 32}{4}. $$Step 3: Verify that the $$k$$ values for both boundaries are the same, as the interval is symmetric around the mean. Use this $$k$$ value in Chebychev's formula $$1 - \frac{1}{k^2} to $$determine the proportion of data within this range. Step 4: Multiply the proportion obtained from Chebychev's formula by the total number of customers (40) to find the minimum number of customers who spent between $$24$$ and $$40. $$Step 5: Interpret the result in the context of the problem, ensuring that the answer represents the minimum number of customers as guaranteed by Chebychev's Theorem.

The mean sale per customer for 40 customers at a gas station is \$32.00, with a standard deviation of \$4.00. Using Chebychev’s Theorem, determine at least how many of the customers spent between \$24.00 and \$40.00.

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Key Concepts

Mean

Standard Deviation

Chebyshev's Theorem