Question #51128

Take a sample of thirty households from two counties A and B and collect data on their incomes for the past month. The information obtained (in thousands of dollars) and their corresponding frequencies is given below:
A's Income Frequency B's Income Frequency
35 4 23 2
49 7 15 4
21 3 12 6
39 5 11 7
15 6 32 5
28 3 18 2
25 2 22 4
i)Calculate the median income of A
ii) Calculate the mode income of B
iii) Based on arithmetic mean income which county has lower average monthly incomes?
iv) Calculate standard deviation of the income distribution of each county.
v) Based on the coefficient of variation which of the counties has least variability in income distribution?

Expert's answer

A's Income Frequency B's Income Frequency

35 4 23 2

49 7 15 4

21 3 12 6

39 5 11 7

15 6 32 5

28 3 18 2

25 2 22 4

i) The median income of A is: Me = 35

ii) The mode income of B is: Mo = 11

iii) The mean income of A is: M = (35*4 + 49*7 + 21*3 + 39*5 + 15*6 + 28*3 + 25*2)/30 = 32.17

The mean income of B is: M = (23*2 + 15*4 + 12*6 + 11*7 + 32*5 + 18*2 + 22*4)/30 = 17.97

Based on arithmetic mean income county B has lower average monthly incomes, than country A.

iv) The standard deviation of the income distribution of each county is:

sA = ((4*(35-32.17)^2+7*(49-32.17)^2+6*(12-17.97)^2+7*(11-17.97)^2+5*(32-17.97)^2+2*(18-17.97)^2+4*(22-17.97)^2)/30)^0.5 = 12.31

sB = ((2*(23-17.97)^2+4*(15-17.97)^2+3*(21-32.17)^2+5*(39-32.17)^2+6*(15-32.17)^2+3*(28-32.17)^2+2*(25-32.17)^2)/30)^0.5 = 10.98

v) Based on the coefficient of variation:

CV = s/m*100%,

CVa = 12.31/32.17*100% = 38.3%

CVb = 10.98/17.97*100% = 61.1%

So, the country A has least variability in income distribution.

35 4 23 2

49 7 15 4

21 3 12 6

39 5 11 7

15 6 32 5

28 3 18 2

25 2 22 4

i) The median income of A is: Me = 35

ii) The mode income of B is: Mo = 11

iii) The mean income of A is: M = (35*4 + 49*7 + 21*3 + 39*5 + 15*6 + 28*3 + 25*2)/30 = 32.17

The mean income of B is: M = (23*2 + 15*4 + 12*6 + 11*7 + 32*5 + 18*2 + 22*4)/30 = 17.97

Based on arithmetic mean income county B has lower average monthly incomes, than country A.

iv) The standard deviation of the income distribution of each county is:

sA = ((4*(35-32.17)^2+7*(49-32.17)^2+6*(12-17.97)^2+7*(11-17.97)^2+5*(32-17.97)^2+2*(18-17.97)^2+4*(22-17.97)^2)/30)^0.5 = 12.31

sB = ((2*(23-17.97)^2+4*(15-17.97)^2+3*(21-32.17)^2+5*(39-32.17)^2+6*(15-32.17)^2+3*(28-32.17)^2+2*(25-32.17)^2)/30)^0.5 = 10.98

v) Based on the coefficient of variation:

CV = s/m*100%,

CVa = 12.31/32.17*100% = 38.3%

CVb = 10.98/17.97*100% = 61.1%

So, the country A has least variability in income distribution.

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