Final answer:
Chebyshev's theorem provides a way to determine the proportion of data that falls within a certain number of standard deviations from the mean. Using this theorem, we can calculate the range that is at least 2.5 standard deviations from the mean.
Step-by-step explanation:
Chebyshev's theorem provides a way to determine the proportion of data that falls within a certain number of standard deviations from the mean. According to Chebyshev's theorem, at least 75 percent of the data falls within two standard deviations of the mean, at least 89 percent falls within three standard deviations, and at least 95 percent falls within 4.5 standard deviations of the mean.
Therefore, if we want to determine the proportion of data that falls within a specific range, such as 2.5 standard deviations from the mean, we can use Chebyshev's theorem to calculate it.
For example, if we have a data set with a mean of 10 and a standard deviation of 2, we can calculate the range that is 2.5 standard deviations from the mean:
Lower limit = mean - (2.5 * standard deviation)
Lower limit = 10 - (2.5 * 2) = 10 - 5 = 5
Upper limit = mean + (2.5 * standard deviation)
Upper limit = 10 + (2.5 * 2) = 10 + 5 = 15
Therefore, at least 2.5 standard deviations of the data in this example falls between 5 and 15.