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?

Construct a​ 95% confidence interval estimate of the percentage of presidents who were taller than their opponents.

PRESIDENT HEIGHT and HEIGHT OPP

Cleveland 180 180

J. Kennedy 183 182

Garfield 183 187

Reagan 185 177

Eisenhower 179 178

Nixon 182 180

Carter 177 183

Harrison 168 180

F. Roosevelt 188 182

Clinton 188 188

Taylor 173 174

Harrison 173 168

Johnson 192 180

Jefferson 189 170

Buchanan 183 175

T. Roosevelt 178 175

J. Adams 170 189

G. H. W. Bush 188 173

Taft 182 178

Van Buren 168 180

Answer 1

From the given data we find that there are 13 out of 20 presidents who were taller than their opponents

13 out of 20 presidents were taller than their opponents       
Thus sample proportion p̂ = 13/20 = 0.65       
       
To find 95% confidence interval for true population proportion       
For 95%, α = 0.05, α/2 = 0.025       
From the z-tables, or Excel function NORM.S.INV(α/2)       
z = NORM.S.INV(0.025) = 1.96   (We take the positive value for calculations)    
Confidence interval is given by        

= (0.441, 0.859)       

95% confidence interval for population proportion presidents taller than their opponents is (0.441, 0.859)       
       

Since the confidence interval contains the proportions 0.5 or less than 0.5, it means that there is a possibility that the proportion of presidents with greater heights than opponents can be less than 50%

This disproves the claim that it appear that greater height is an advantage for presidential​ candidates.