Question

How Democratic is Georgia County by-county results are shown for a recent election. For your convenience, the dates have been sorted in increasing order (Source County and City Data Hook, 12th edition, U.S. Census Bureau). Percentage of Democratic Vote by Counties in George 31 33 34 34 35 35 35 36 38 40 41 41 41 41 41 41 41 45 46 46 46 46 47 48 49 49 53 53 53 53 53 55 56 56 57 57 (a) Select the box-and-whisker plot of the data

Answer 1

a) Box-and-whisker plot: plot 1st

b)

5 point summary:

Minimum = 31

First quartile, Q1 = 40

Median =45

Third quartile, Q3 = 52.5

Maximum = 68

Interquartile range, IQR = Q3 - Q1 = 52.5 - 40 = 12.5

c)

Class interval	Frequency
31 - 38	11
39 - 46	24
47 - 54	15
55 - 62	7
63 - 70	3
Total	60

Class interval	Midpoint, x	Frequency, f	fx	fx²
31 - 38	34.5	11	379.5	13092.75
39 - 46	42.5	24	1020	43350
47 - 54	50.5	15	757.5	38253.75
55 - 62	58.5	7	409.5	23955.75
63 - 70	66.5	3	199.5	13266.75
Total		60	2766	131919

Mean, x̅ = ∑fx / n = 2766/60 = 46.1

Standard deviation, s = √((∑fx² - (∑fx)²/n)/(n-1)) = √((131919 - 2766²/60)/(60- 1))

= 8.6420 = 8.64

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Chebyshev's theorem states that the proportion or percentage of any data set that lies within k standard deviation of the mean where k is any positive integer greater than 1 is at least 1 – 1/k^2.

0.75 = 1-1/k²

1/k² = 1 - 0.75

k² = 1/0.25

k = 2

75% Chebyshev interval centered about the mean.

Lower limit = x̅ - 2*s = 46.1 - 2*8.64 = 28.82

Upper imit = x̅ + 2*s = 46.1 + 2*8.64 = 63.38

d)

∑x = 2769

∑x² = 132179

n = 60

Mean, x̅ = Ʃx/n = 2769/60 = 46.15

Standard deviation, s = √[(Ʃx² - (Ʃx)²/n)/(n-1)] = √[(132179-(2769)²/60)/(60-1)]

= 8.6256 = 8.63