The Russian mathmatician Pafnuti L. Chebychev pved a remarkable theorem called Chebybchev's Inequality: In any data set, if p is the fraction of the data that lies within k standard deviations to either side of the mean, then p is greater than or equal to 1-1/k^2.
a. According to Chebychev's Theorem, what percent of a data set must lie within 2 standard deviations of the mean?
b. What percent must lie within 3 standard deviations?
To determine the percentage of a data set lying within a certain number of standard deviations from the mean using Chebyshev's Inequality, you can follow these steps:
a. According to Chebyshev's Theorem, if k = 2, then we need to find p when k = 2.
1. Substitute k = 2 into the inequality: p ≥ 1 - 1/k^2.
p ≥ 1 - 1/2^2.
2. Calculate 1/2^2: 1/4.
3. Simplify the inequality to obtain: p ≥ 1 - 1/4.
4. Calculate 1 - 1/4: 3/4.
Thus, at least 3/4 (or 75%) of the data set must lie within 2 standard deviations of the mean.
b. Similarly, if k = 3, we need to find p when k = 3.
1. Substitute k = 3 into the inequality: p ≥ 1 - 1/k^2.
p ≥ 1 - 1/3^2.
2. Calculate 1/3^2: 1/9.
3. Simplify the inequality to obtain: p ≥ 1 - 1/9.
4. Calculate 1 - 1/9: 8/9.
Thus, at least 8/9 (or approximately 88.89%) of the data set must lie within 3 standard deviations of the mean.