Question

Listed below are the heights of candidates who won elections and the heights of the candidates with the next highest number of votes. The data are in chronological​ order, so the corresponding heights from the two lists are matched. Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal. Construct a​ 95% confidence interval estimate of the mean of the population of all​ "winner/runner-up" differences. Does height appear to be an important factor in winning an​ election? Winner 71 69 71 69 70 72 73 74 ​Runner-Up 69 71 68 68 68 69 68 73

Answer 1

paired test

Sample #1	Sample #2	difference , Di =sample1-sample2	(Di - Dbar)²
71	69	2	0.0156
69	71	-2	15.0156
71	68	3	1.2656
69	68	1	0.7656
70	68	2	0.0156
72	69	3	1.2656
73	68	5	9.7656
74	73	1.0000	0.7656

	sample 1	sample 2	Di	(Di - Dbar)²
sum =	569	554	15	28.875

mean of difference ,    D̅ =ΣDi / n =   1.8750
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    2.0310

sample size ,    n =    8          
Degree of freedom, DF=   n - 1 =    7   and α =    0.05  
t-critical value =    t α/2,df =    2.3646   [excel function: =t.inv.2t(α/2,df) ]      
                  
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    2.0310          
                  
std error , SE = Sd / √n =    2.0310   / √   8   =   0.7181
margin of error, E = t*SE =    2.3646   *   0.7181   =   1.6980
                  
mean of difference ,    D̅ =   1.875          
confidence interval is                   
Interval Lower Limit= D̅ - E =   1.875   -   1.6980   =   0.1770
Interval Upper Limit= D̅ + E =   1.875   +   1.6980   =   3.5730
                  
so, confidence interval is (   0.1770   < Dbar <   3.5730   )