Question

Refer to the data set of 20 randomly selected presidents given below. Treat the data as a sample and find the proportion of presidents who were taller than their opponents. Use that result to construct a​ 95% confidence interval estimate of the population percentage. Based on the​ result, does it appear that greater height is an advantage for presidential​ candidates? Why or why​ not?

Click the icon to view the table of heights.

Construct a​ 95% confidence interval estimate of the percentage of presidents who were taller than their opponents. nothing​%less than p less than nothing​% ​(Round to one decimal place as​ needed.)

Greater height ▼ (appears or does not appear) to be an advantage for presidential candidates. If greater height was an​ advantage, then taller candidates should have won ▼ (exactly, less than, or more than) ​50% of the elections and the confidence interval ▼ (does not or does) include​ 50%.

This is the chart

PRESIDENT	HEIGHT	OPP
Harrison	168	180
Harding	183	178
Lincoln	193	188
Wilson	180	182
Jackson	185	171
Truman	175	173
Hoover	182	180
Coolidge	178	180
Jefferson	189	170
Eisenhower	179	178
Clinton	188	188
G. W. Bush	183	185
Polk	173	185
Taylor	173	174
Taft	182	178
Hayes	173	178
Nixon	182	180
Reagan	185	177
Cleveland	180	180
Johnson	192	180

Answer 1

Let d = height - opp

If president height is greater than opp then d must be positive. Following table shows the d:

PRESIDENT	HEIGHT	OPP	d=height-opp
Harrison	168	180	-12
Harding	183	178	5
Lincoln	193	188	5
Wilson	180	182	-2
Jackson	185	171	14
Truman	175	173	2
Hoover	182	180	2
Coolidge	178	180	-2
Jefferson	189	170	19
Eisenhower	179	178	1
Clinton	188	188	0
G. W. Bush	183	185	-2
Polk	173	185	-12
Taylor	173	174	-1
Taft	182	178	4
Hayes	173	178	-5
Nixon	182	180	2
Reagan	185	177	8
Cleveland	180	180	0
Johnson	192	180	12

Out of 20 d's. 13 are positive. So the required confidence interval is

Hence, the required confidence interval is

44.1% < p < 85.9%

Greater height does not appear to be an advantage for presidential candidates. If greater height was an​ advantage, then taller candidates should have won more than ​50% of the elections and the confidence interval does not include​ 50%.