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?
F. Roosevelt 188 182

Harding 183 178

Polk 173 185

Clinton 188 188

J. Adams 170 189

Truman 175 173

J. Q. Adams 171 191

Eisenhower 179 178

Harrison 168 180

G. H. W. Bush 188 173

Carter 177 183

T. Roosevelt 178 175

Hayes 173 178

Buchanan 183 175

Taylor 173 174

Taft 182 178

Harrison 173 168

Hoover 182 180

Coolidge 178 180

Jackson 185 171

1.) ___% < p < ___%

Answer 1

Answer:

1.) 33% < p < 66%

As the confidence interval includes values under 50% and over 50%, it doesn't appear that greater height is an advantage for presidential.

If the lower bound of the confidence interval were over 50%, one could interpret that greater height is an advantage for presidential, but it is not the case for this sample.

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Step-by-step explanation:

Answer 2

Answer:

1.) 33% < p < 66%

As the confidence interval includes values under 50% and over 50%, it doesn't appear that greater height is an advantage for presidential.

If the lower bound of the confidence interval were over 50%, one could interpret that greater height is an advantage for presidential, but it is not the case for this sample.

Step-by-step explanation:

Out of this sample, we have 11 presidents, out of 20, that were taller than their oponent.

Then, the proportion of presindents that were taller than their oponent can be calculated as:

$p=X/n=11/20=0.55$

We can calculate now the standard error of the proportion as:

$\sigma_p=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.55*0.45}{20}}=\sqrt{0.012375}=0.11$

For a 95% confidence interval, the z-value is z=1.96 (we can loook up this value in the standarized normal distribution table).

Then, the lower and upper bounds of the confidence interval are:

$LL=p-z\cdot \sigma_p=0.55-1.96*0.11=0.55-0.22=0.33\\\\UL=p+z\cdot \sigma_p=0.55+1.96*0.11=0.55+0.22=0.66$

As the confidence interval includes values under 50% and over 50%, it doesn't appear that greater height is an advantage for presidential.

If the lower bound of the confidence interval were over 50%, one could interpret that greater height is an advantage for presidential, but it is not the case for this sample.