From a sample with $n=32$, the mean duration of a geyser's eruptions is 3.01 minutes and the standard deviation is 0.59 minutes. Using Chebychev's Theorem, determine at least how many of the eruptions lasted between 1.83 and 4.19 minutes.
At least of the eruptions lasted between 1.83 and 4.19 minutes. (Simplify your answer.)
Answer
Rounding off to the nearest whole number, we get the final answer. Therefore, at least \(\boxed{24}\) eruptions lasted between 1.83 and 4.19 minutes.
Step 1 ：Given a sample with \(n=32\), the mean duration of a geyser's eruptions is 3.01 minutes and the standard deviation is 0.59 minutes. We are asked to determine at least how many of the eruptions lasted between 1.83 and 4.19 minutes.
Step 2 ：We can use Chebyshev's theorem which states that at least \(1 - \frac{1}{k^2}\) of data from a sample will fall within \(k\) standard deviations from the mean, where \(k\) is any positive real number greater than 1.
Step 3 ：First, we need to find the value of \(k\) for both the lower and upper bounds of the duration. The lower bound \(k\) value (\(k1\)) can be calculated as \(\frac{mean - lower bound}{standard deviation}\) and the upper bound \(k\) value (\(k2\)) can be calculated as \(\frac{upper bound - mean}{standard deviation}\).
Step 4 ：Calculating for \(k1\) and \(k2\), we get \(k1 = 1.9999999999999996\) and \(k2 = 2.0000000000000013\).
Step 5 ：Then, we need to find the minimum \(k\) value between \(k1\) and \(k2\) because we want to find the maximum proportion of data that falls within these \(k\) standard deviations from the mean. In this case, \(k = 1.9999999999999996\).
Step 6 ：Finally, we can calculate the number of eruptions that lasted between 1.83 and 4.19 minutes by multiplying the proportion of data that falls within these \(k\) standard deviations from the mean by the total number of data points (\(n\)). The proportion is calculated as \(1 - \frac{1}{k^2}\), which gives us a proportion of 0.7499999999999999.
Step 7 ：Multiplying the proportion by the total number of data points, we get the number of eruptions as 23.999999999999996.
Step 8 ：Rounding off to the nearest whole number, we get the final answer. Therefore, at least \(\boxed{24}\) eruptions lasted between 1.83 and 4.19 minutes.
