Question

How Democratic is Georgia? County-by-county results are shown for a recent election. For your convenience, the data have been sorted in increasing order (Source: County and City Data Book, 12th edition, U.S. Census Bureau). 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

Then estimate the mean and sample standard deviation using the frequency table. (Round x to one decimal place, and round s to two decimal places.)

Compute a 75% Chebyshev interval centered about the mean. (Round your answers to two decimal places.)

Lower Limit :28.82

upper Limit: 63.38

(d) If you have a statistical calculator or computer, use it to find the actual sample mean and sample standard deviation. Otherwise, use the values Σx = 2769 and Σx² = 132,179 to compute the sample mean and sample standard deviation. (Round your answers to two decimal places.)??

Answer 1

sample measn = = 2769/60=46.2

and sample s.d. = s= = 8.63

According to Chebyshev's Theorem,
[1 - (1/k^2)] proportion of values will fall between Mean +/- (k*SD)
k in this case equal to z
z = (X-Mean)/SD
X = Mean + (z*SD)
So,
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

Therefore, z = k = 2

Required Interval is:
Mean - (z * SD) < X < Mean + (z * SD)
46.2- (2 * 8.63) < X < 46.2 - (2 * 8.63)
46.2 - 17.26 < X < 46.2 + 17.26
28.94< X <63.46