e Chebyshev’s theorem to find what percent of the values will fall between 226 and 340 for a data set with a mean of 283 and standard deviation of 19.
To use Chebyshev's theorem to find the percentage of values that fall between a certain range, you need to calculate the number of standard deviations away from the mean that the range corresponds to. Then, you can use the Chebyshev's theorem formula to find the minimum percentage of values that fall within that range.
Chebyshev's theorem states that for any data set, regardless of its distribution, at least (1 - 1/k^2) of the values will fall within k standard deviations from the mean, where k is any positive number greater than 1.
First, let's calculate the number of standard deviations away from the mean that the given range corresponds to:
For the lower bound of 226:
Distance from the mean = (226 - 283) / 19 ≈ -3
For the upper bound of 340:
Distance from the mean = (340 - 283) / 19 ≈ 3
Next, substitute the absolute value of the distance (since the theorem is about the number of deviations from the mean) into Chebyshev's theorem formula:
Percentage of values within the given range = 1 - 1/k^2
Since the range corresponds to 3 standard deviations from the mean, we have k = 3.
Percentage of values within the range = 1 - 1/3^2
= 1 - 1/9
= 8/9
≈ 0.8889
Therefore, according to Chebyshev's theorem, at least 88.89% of the values will fall between 226 and 340 for a data set with a mean of 283 and a standard deviation of 19.