Question

y. Su l e ule results. 112. The masses (in kilograms) of a sample of five boxes being sent by United Parcel Service (UPS) are: 12, 6, 7, 3, and 10. a. Compute the range. b. Compute the mean deviation. c. Compute the standard deviation d. Using Chebyshev's theorem, at least what percentage of the observations must be within 2.5 standard deviations of the mean? Verify. e. Using the Empirical Rule, about 68% of the values would occur between what values? Verify.

Answer 1

3, 6, 7, 10, 12

a) range = 12 - 3 = 9

b) mean = (3 + 6 + 7 + 10 + 12)/5 = 7.6

mean deviation = ((3 - 7.6) + (6 - 7.6) + (7 - 7.6) + (10 - 7.6) + (12 - 7.6))/5

                         = ((-4.6) + (-1.6) + (-0.6) + 2.4 + 4.4)/5 = 0

c) standard deviation = sqrt(((-4.6)^2 + (-1.6)^2 + (-0.6)^2 + (2.4)^2 + (4.4)^2)/4) = 3.507

d) k = 2.5

1 - 1/k²

= 1 - 1/(2.5)^2

= 1 - 0.16

= 0.84 = 84%

According to Chebyshev's theorem at least 84% of the observations must be within 2.5 standard deviations of the mean.

e) According to the empirical about 68% of the data fall within one standard deviation from the mean.

mean - 2 * SD = 7.6 - 2 * 3.507 = 0.586

mean + 2 * SD = 7.6 + 2 * 3.507 = 14.614

So the two values are 0.586 and 14.614