A data set has a mean of 1500 and a standard deviation of 80

A data set has a mean of 1,500 and a standard deviation of 80.

Using Chebyshev\'s theorem, what percentage of the observations fall between 1,340 and 1,660? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)

Using Chebyshev’s theorem, what percentage of the observations fall between 1,180 and 1,820? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)

Solution

According to Chebyshev\'s theorem, at least 1 - 1/k^2 of the data is within k standard deviations from the mean.

Thus,

a)

Note that

k = (x - u)/s

Thus,

1340 and 1660 are 2 standard deviations from 1500, as

k = (1660 - 1500)/80 = 2

k = (1340 - 1500)/80 = -2

Thus,

1 - 1/k^2 = 1 - 1/2^2 = 0.75 = 75% [answer]

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b)

Note that

k = (x - u)/s

Thus,

1340 and 1660 are 2 standard deviations from 1500, as

k = (1180 - 1500)/80 = -4

k = (1820 - 1500)/80 = 4

Thus,

1 - 1/k^2 = 1 - 1/4^2 = 0.9375 = 93.75% = 94% [answer]