Question

Consider a sample with a mean of 50 and a standard deviation of 4. Use Chebyshev's theorem to determine the percentage of the data within each of the following ranges (to the nearest whole number).

a. 30 to 70, at least %
b. 35 to 65, at least %
c. 41 to 59, at least %
d. 38 to 62, at least %
e. 33 to 67, at least %

Answer 1

a)
`z = (30-50)/(4)= -5`

`z = (70-50)/(4)= 5`

`1- (1)/(5^2) = 0.96 = 96\%`

b)
`z = (35-50)/(4)= -3.75`

`z = (65-50)/(4)= 3.75`

`1- (1)/(3.75^2) = 0.929 = 92.9\%`

c)
`z = (41-50)/(4)= -2.25`

`z = (59-50)/(4)= 2.25`

`1- (1)/(2.25^2) = 0.8025 = 80.25\%`

d)
`z = (38-50)/(4)= -3`

`z = (62-50)/(4)= 3`

`1- (1)/(3^2) = 0.889 = 88.89\%`

e)
`z = (33-50)/(4)= -4.25`

`z = (67-50)/(4)= 4.25`

`1- (1)/(4.25^2) = 0.9446 = 94.46\%`

Explanation:

Data given

`\mu =50` reprsent the population mean

`\sigma=4` represent the population standard deviation

The Chebyshev's Theorem states that for any dataset

• We have at least 75% of all the data within two deviations from the mean.

• We have at least 88.9% of all the data within three deviations from the mean.

• We have at least 93.8% of all the data within four deviations from the mean.

Or in general words "For any set of data (either population or sample) and for any constant k greater than 1, the proportion of the data that must lie within k standard deviations on either side of the mean is at least:
`1-(1)/(k^2)`

Part a

For this case we can find the deviations from the mean like this:

`z = (30-50)/(4)= -5`

`z = (70-50)/(4)= 5`

`1- (1)/(5^2) = 0.96 = 96\%`

Part b

For this case we can find the deviations from the mean like this:

`z = (35-50)/(4)= -3.75`

`z = (65-50)/(4)= 3.75`

`1- (1)/(3.75^2) = 0.929 = 92.9\%`

Part c

For this case we can find the deviations from the mean like this:

`z = (41-50)/(4)= -2.25`

`z = (59-50)/(4)= 2.25`

`1- (1)/(2.25^2) = 0.8025 = 80.25\%`

Part d

For this case we can find the deviations from the mean like this:

`z = (38-50)/(4)= -3`

`z = (62-50)/(4)= 3`

`1- (1)/(3^2) = 0.889 = 88.89\%`

Part e

For this case we can find the deviations from the mean like this:

`z = (33-50)/(4)= -4.25`

`z = (67-50)/(4)= 4.25`

`1- (1)/(4.25^2) = 0.9446 = 94.46\%`