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 75 72 73 74 72 73 76 73

​Runner-Up 74 71 70 70 69 73 72 72

Construct the​ 95% confidence interval.​ (Subtract the height of the​ runner-up from the height of the winner to find the​ difference, d.)

B) Based on the confidence interval, does the height appear to be an important factor in winning an election?

Answer 1

The below is the table of difference between two variables.

Winner(x)	Runner-Up(y)	d =(x -y)	(d-dbar)^2
75	74	1	1.265625
72	71	1	1.265625
73	70	3	0.765625
74	70	4	3.515625
72	69	3	0.765625
73	73	0	4.515625
76	72	4	3.515625
73	72	1	1.265625
Total		17	16.875

From the above table, = 17/8 = 2.125

S_D² = 16.875 /7 = 2.4107

S_D = 1.5526   n = 8

The 95% confidence interval is given by ± t_α/2 (s_D / √n)

The t critical value t_α/2,n-1 = t_0.025,7 = 2.365

95% confidence interval is given by

± t_0.025 (s_d / √n) = (2.125 ± 2.365*1.553/sqrt(8)) = (0.826, 3.423)

The 95% confidence interval for the mean difference μ_D is 0.826 to 3.423.

Since average difference 2.125 is within the confidence intervals, so we can say that there is not the significant average difference in the height.

As a result, height does not appear to be an important factor in winning an election.