Chebychev's Theorem: Old Faithful Geyser at Yellowstone

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In summary, Chebychev's Theorem, also known as the Chebychev Inequality, is a mathematical principle that can be used to predict the likelihood of an event occurring within a certain time frame based on past data. It can be applied to Old Faithful Geyser at Yellowstone by analyzing the time intervals between its eruptions. However, there are limitations to using this method, such as assuming a normal distribution of data and not accounting for potential changes in the geyser's behavior. It can also be applied to other geysers or natural phenomena, but should be used with caution and in conjunction with other methods and observations.
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Homework Statement



Chebychev's Theorem: Old Faithful is a famous geyser at Yellowstone National
Park. From a sample with n = 32, the mean duration of Old Faithful's eruptions is
3.32 minutes and the standard deviation is 1.09 minutes. Using the Chebychev's
Theorem, determine at least how many of the eruptions lasted between 1.14
minutes and 5.5 minutes.

The Book gives 24 as an answer...but I can't figure out how to come up with that.
 
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How many standard deviations is the data set bound that they want you to account for?
 
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I can confirm that Chebychev's Theorem is a mathematical principle that can be applied to any set of data, including the duration of Old Faithful's eruptions. This theorem states that for any given set of data, regardless of its distribution, a certain percentage of the data will fall within a certain number of standard deviations from the mean. In this case, we can use Chebychev's Theorem to determine the percentage of Old Faithful's eruptions that fall between 1.14 minutes and 5.5 minutes, given the mean and standard deviation provided.

To use Chebychev's Theorem, we first need to calculate the number of standard deviations that the range of 1.14 minutes to 5.5 minutes falls within. This can be done by subtracting the mean (3.32 minutes) from each value and then dividing by the standard deviation (1.09 minutes). This gives us a range of -1.18 to 2.47 standard deviations.

Next, we need to determine the percentage of data that falls within this range. According to Chebychev's Theorem, at least 75% of the data will fall within 2 standard deviations from the mean. This means that at least 75% of Old Faithful's eruptions will fall within the range of 1.14 minutes to 5.5 minutes.

To determine the minimum number of eruptions that fall within this range, we can multiply the total number of eruptions (32) by 75%, giving us 24 eruptions. This means that at least 24 of the 32 eruptions will last between 1.14 minutes and 5.5 minutes.

In conclusion, Chebychev's Theorem can be used to determine the minimum number of Old Faithful's eruptions that will last between 1.14 minutes and 5.5 minutes, based on the mean and standard deviation provided. However, it is important to note that this is a minimum estimate and the actual number of eruptions within this range may be higher.
 

Related to Chebychev's Theorem: Old Faithful Geyser at Yellowstone

1. What is Chebychev's Theorem and how does it apply to Old Faithful Geyser at Yellowstone?

Chebychev's Theorem, also known as the Chebychev Inequality, is a mathematical principle that states that for any data set, the proportion of data points that lie within a certain number of standard deviations from the mean is at least 1-1/k^2, where k is the number of standard deviations. In the case of Old Faithful Geyser, Chebychev's Theorem can be used to predict the likelihood of the geyser erupting within a certain time frame based on its past eruption data.

2. How is Chebychev's Theorem used to make predictions about Old Faithful Geyser?

Chebychev's Theorem can be used to make predictions about Old Faithful Geyser by analyzing the time intervals between its eruptions. By calculating the mean and standard deviation of these intervals, we can use Chebychev's Theorem to determine the proportion of eruptions that are likely to occur within a certain time frame. This allows us to make more accurate predictions about when the geyser will erupt.

3. Is Chebychev's Theorem a reliable method for predicting the eruption of Old Faithful Geyser?

Chebychev's Theorem is a statistical principle that provides a general prediction based on past data. While it can give us a rough estimate of when Old Faithful Geyser will erupt, it cannot account for any potential changes or anomalies in the geyser's behavior. Therefore, it should not be relied upon as the sole method for predicting eruptions and should be used in conjunction with other methods and observations.

4. What are some limitations of using Chebychev's Theorem for predicting Old Faithful Geyser's eruptions?

One limitation of using Chebychev's Theorem for predicting Old Faithful Geyser's eruptions is that it assumes a normal distribution of data, which may not always be the case. Additionally, the theorem does not take into account any potential changes in the geyser's behavior, such as changes in its underground plumbing system or external factors like earthquakes. It is important to also consider other factors and observations when predicting geyser eruptions.

5. Can Chebychev's Theorem be applied to other geysers or natural phenomena?

Yes, Chebychev's Theorem can be applied to other geysers or natural phenomena that exhibit a certain level of regularity or periodicity in their behavior. It can also be applied to a wide range of data sets in various fields, such as finance, economics, and biology, to make predictions and identify outliers. However, as with any statistical principle, it should be used with caution and in conjunction with other methods and observations.

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