Using Chebychev’s Theorem Old Faithful is a famous geyser at Yellowstone National Park. From a sample with n = 100, the mean interval between Old Faithful’s eruptions is 101.56 minutes and the standard deviation is 42.69 minutes. Using Chebychev’s Theorem, determine at least how many of the intervals lasted between 16.18 minutes and 186.94 minutes. (Adapted from Geyser Times)

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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 \(k > 1\).

Step 2: Calculate the number of standard deviations \(k\) that the given interval \([16.18, 186.94]\) spans from the mean. Use the formula \(k = \frac{|x - \mu|}{\sigma}\), where \(x\) is the boundary value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Compute \(k\) for both boundaries: \(k = \frac{101.56 - 16.18}{42.69}\) and \(k = \frac{186.94 - 101.56}{42.69}\).

Step 3: Verify that the calculated \(k\) values for both boundaries are approximately equal (they should be, as the interval is symmetric around the mean). Use the larger \(k\) value for the next step.

Step 4: Apply Chebychev's Theorem to determine the proportion of data within \(k\) standard deviations. Substitute \(k\) into the formula \(1 - \frac{1}{k^2}\) to find the minimum proportion of data within the interval.

Step 5: Multiply the proportion obtained in Step 4 by the total sample size \(n = 100\) to determine the minimum number of intervals that lasted between 16.18 minutes and 186.94 minutes.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chebyshev's Theorem

Chebyshev's Theorem states that for any dataset, regardless of its distribution, at least 1 - (1/k²) of the data values will fall within k standard deviations from the mean. This theorem is particularly useful for understanding the spread of data and making inferences about the proportion of values that lie within a certain range, especially when the distribution is unknown.

Mean and Standard Deviation

The mean is the average of a set of values, calculated by summing all values and dividing by the number of values. The standard deviation measures the amount of variation or dispersion in a set of values, indicating how much individual data points deviate from the mean. Together, these statistics provide a summary of the data's central tendency and variability.

Interval Calculation

In the context of Chebyshev's Theorem, calculating the interval involves determining how many standard deviations the specified range (16.18 to 186.94 minutes) is from the mean (101.56 minutes). This calculation helps in applying the theorem to find the minimum proportion of data points that fall within this interval, allowing for a better understanding of the distribution of eruption intervals.

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