Answer:
Step-by-step explanation:
Corresponding heights of presidents and height of their main opponents form matched pairs.
The data for the test are the differences between the heights.
μd = the president's height minus their main opponent's height.
President's height. main opp diff
191. 166. 25
180. 179. 1
180. 168. 12
182. 183. - 1
197. 194. 3
180. 186. - 6
Sample mean, xd
= (25 + 1 + 12 - 1 + 3 + 6)/6 = 5.67
xd = 5.67
Standard deviation = √(summation(x - mean)²/n
n = 6
Summation(x - mean)² = (25 - 5.67)^2 + (1 - 5.67)^2 + (12 - 5.67)^2+ (- 1 - 5.67)^2 + (3 - 5.67)^2 + (- 6 - 5.67)^2 = 623.3334
Standard deviation = √(623.3334/6 sd = 10.19
For the null hypothesis
H0: μd ≥ 0
For the alternative hypothesis
H1: μd < 0
The distribution is a students t. Therefore, degree of freedom, df = n - 1 = 6 - 1 = 5
The formula for determining the test statistic is
t = (xd - μd)/(sd/√n)
t = (5.67 - 0)/(10.19/√6)
t = 1.36
We would determine the probability value by using the t test calculator.
p = 0.12
Since alpha, 0.05 < than the p value, 0.12, then we would fail to reject the null hypothesis.
Therefore, at 5% significance level, we can conclude that for the population of heights for presidents and their main opponents, the differences have a mean greater than 0 cm.
and plug in r=4 and h=13 to get 
, so 
.