How Democratic is Georgia? County by- county results are Book, 12th edition, U.S

Question

How Democratic is Georgia? County by- county results are Book, 12th edition, U.S. Census Bureau). r shown for a recent election. For your convenience, the data have been sorted in increasing order (Souroe: County and City Dat Percentage of Democratic Vote by Counties in Georgia 31 33 34 34 35 35 35 36 38 38 38 39 40 40 40 40 41 41 41 41 41 41 41 42 42 43 44 44 44 45 45 46 46 46 46 47 48 49 49 49 49 50 51 52 52 53 53 53 53 53 55 56 56 57 57 59 62 66 66 68 (a) Select the box-and-whisker plot of the data. 1020304, 506070

Explanation / Answer

Kindly note , have answered only those questions which you have got wrong

b) Interquartile range : Q3 - Q1

Sol : We first find median of the data .

Median = M = N / 2 { We have considered N / 2 as we have total number of observations even i.e 60}

M = 60 / 2 = 30th observation

M = 45

So we divide the data in two parts ,1st part above 30th observation and 2nd part below 30th observation .

The 1st part is called Q1 which has 29 observations and Median of Q1 = ( N + 1 ) / 2 { as N=29 is odd}

therefore Q1 = 30 / 2 = 15th observation = 40

The 2nd part is called Q3 which has 30 observations and Median of Q3 = N / 2 { as N=30 is even}

therefore Q3 = 30 / 2 = 15th observation = 52

Now , interquartile range IQR = Q3 - Q1 = 52 - 40 = 12

IQR = 12.

c) Compute 75% chebyshec interval centered about mean .

According to Chebyshev's Theorem,
[1 - (1/k^2)] proportion of values will fall between Mean +/- (k*SD)

1 - 1/k^2 = 0.75
- 1/k^2 = 0.75 - 1= - 0.25
1/k^2 = 0.25
k^2 = 1/0.25
k^2 = 4
k = 2

therfore Mean - ( k * SD ) = 46.15 - ( 2* 8.63) = 28.90 = Lower limit

Mean + (k * SD ) = 46.15 + ( 2* 8.63 ) = 63.40 = Upper limit

d) Compute Standard Deviation

By using Excel formula STDEV we get s = 8.63 where the formula is sqrt ( x - mean ) ^ 2 / n-1

But in statistical calculator STDEV we get s = 8.55 where we have choice of selecting denominator sqrt ( x - mean ) ^ 2 / n-1

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