use chebyshev's theorem to find what percent of the values will fall between 241 and 349 for a data set with a mean of 295 and standard deviation of 18
To use Chebyshev's theorem to find the percent of values that will fall between 241 and 349 for a dataset with a mean of 295 and a standard deviation of 18, follow these steps:
Step 1: Determine the range.
The range of values for which we want to find the percentage is from 241 to 349.
Step 2: Calculate the difference between the mean and the range boundaries.
To calculate the difference between the mean (295) and the lower boundary (241), subtract 241 from 295: 295 - 241 = 54.
Similarly, for the upper boundary (349), subtract 349 from 295: 295 - 349 = -54.
Note: In Chebyshev's theorem, we use the absolute value of the difference between the mean and the boundaries.
Step 3: Calculate the number of standard deviations.
To find the number of standard deviations, divide the absolute difference obtained in step 2 by the standard deviation (18):
Lower boundary: 54 / 18 = 3
Upper boundary: -54 / 18 = -3
Note: The standard deviation is always positive. Taking the absolute difference gives us the number of standard deviations from the mean.
Step 4: Determine the percentage using Chebyshev's theorem.
Chebyshev's theorem states that at least (1 - 1 / k^2) * 100% of the values will fall within k standard deviations from the mean, where k is any positive number greater than 1.
For this question, we can use k = 3 because the range boundaries are 3 standard deviations away from the mean (3 and -3 standard deviations).
Using the equation, the percentage of values falling between 241 and 349 is:
(1 - 1 / 3^2) * 100% = (1 - 1 / 9) * 100% = 88.9%
So, according to Chebyshev's theorem, at least 88.9% of the values fall between 241 and 349 for the given dataset.