a. According to Chebyshev's theorem, for any number of standard deviations k:
- At least (1 - 1/k^2) * 100% of the observations fall within k standard deviations of the mean.
In this case, we want to find the percentage of observations that fall between 1,200 and 1,840, which is a range of 640 units. The mean is 1,520 and the standard deviation is 80.
The distance between the mean and 1,200 is 1,520 - 1,200 = 320 units, which is 320/80 = 4 standard deviations away from the mean.
The distance between the mean and 1,840 is 1,840 - 1,520 = 320 units, which is 320/80 = 4 standard deviations away from the mean.
Therefore, according to Chebyshev's theorem, at least (1 - 1/4^2) * 100% = (1 - 1/16) * 100% = 15/16 * 100% = 93.75% of the observations fall between 1,200 and 1,840.
Rounded to the nearest whole percent, the answer is 94%.
b. Similarly, we want to find the percentage of observations that fall between 1,360 and 1,680, which is a range of 320 units. The mean is still 1,520 and the standard deviation is 80.
The distance between the mean and 1,360 is 1,520 - 1,360 = 160 units, which is 160/80 = 2 standard deviations away from the mean.
The distance between the mean and 1,680 is 1,680 - 1,520 = 160 units, which is 160/80 = 2 standard deviations away from the mean.
Therefore, according to Chebyshev's theorem, at least (1 - 1/2^2) * 100% = (1 - 1/4) * 100% = 3/4 * 100% = 75% of the observations fall between 1,360 and 1,680.
Rounded to the nearest whole percent, the answer is 75%.
A variable has a mean of 1,520 and a standard deviation of 80.
a. Using Chebyshev's theorem, what percentage of the observations fall between 1,200 and 1,840? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)
b. Using Chebyshev’s theorem, what percentage of the observations fall between 1,360 and 1,680? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)
1 answer