Answer:
Chebyshev's Theorem provides a way to determine the minimum proportion of data that falls within a certain range for any data distribution, regardless of its shape. It states that for any continuous distribution, at least (1 - 1/k^2) of the data falls within k standard deviations of the mean, where k is any positive constant greater than 1.
In your case, you want to determine at least how many of the eruptions (out of 36) fall between 0.8 and 5.48 minutes. To do this, you can use Chebyshev's Theorem with k standard deviations.
First, you need to calculate the value of k. You want to include the interval from 0.8 to 5.48 minutes, so:
Upper Limit = 5.48 minutes
Lower Limit = 0.8 minutes
Mean = 3.14 minutes
Standard Deviation = 0.78 minutes
k is the number of standard deviations needed to cover the entire range, so:
k = (Upper Limit - Mean) / Standard Deviation
k = (5.48 - 3.14) / 0.78
k = 2.34 / 0.78
k = 3
So, you need to consider 3 standard deviations to cover the entire range from 0.8 to 5.48 minutes.
Now, you can use Chebyshev's Theorem to find the minimum proportion of data within this range:
At least (1 - 1/k^2) of the data falls within k standard deviations of the mean.
At least (1 - 1/3^2) of the data falls within 1/9 of the mean.
1 - 1/9 = 8/9
So, at least 8/9 of the data falls within the range from 0.8 to 5.48 minutes. To find out how many out of the 36 eruptions fall within this range, you can calculate:
(8/9) * 36 ≈ 32
So, at least 32 of the eruptions fall between 0.8 and 5.48 minutes based on Chebyshev's Theorem.
Explanation: