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.)
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%.
To solve these problems using Chebyshev's theorem, we need to use the following formula:
P(|X - μ| ≤ kσ) ≥ 1 - 1/k^2
Where P represents the percentage of observations, X is the value of the variable, μ is the mean, σ is the standard deviation, and k is the number of standard deviations away from the mean.
a. To find the percentage of observations between 1,200 and 1,840, we need to calculate the number of standard deviations away from the mean for both values.
For 1,200:
|1,200 - 1,520| / 80 = 320 / 80 = 4
For 1,840:
|1,840 - 1,520| / 80 = 320 / 80 = 4
Next, we substitute k = 4 into the formula:
P(|X - μ| ≤ 4σ) ≥ 1 - 1/4^2
P(|X - μ| ≤ 4σ) ≥ 1 - 1/16
P(|X - μ| ≤ 4σ) ≥ 15/16
The percentage of observations between 1,200 and 1,840 is at least 15/16, or 93.75%. Rounding to the nearest whole percent gives us 94%.
b. To find the percentage of observations between 1,360 and 1,680, we follow the same steps.
For 1,360:
|1,360 - 1,520| / 80 = 160 / 80 = 2
For 1,680:
|1,680 - 1,520| / 80 = 160 / 80 = 2
Substituting k = 2 into the formula:
P(|X - μ| ≤ 2σ) ≥ 1 - 1/2^2
P(|X - μ| ≤ 2σ) ≥ 1 - 1/4
P(|X - μ| ≤ 2σ) ≥ 3/4
The percentage of observations between 1,360 and 1,680 is at least 3/4, or 75%.