A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights​ (in centimeters) of randomly selected presidents along with the heights of their main opponents. Complete parts​ (a) and​ (b) below.   Height (cm) of President 178 183 182 180 200 179   Height (cm) of Main Opponent 170 189 179 180 195 172   a. Use the sample data with a 0.01 significance level to test the claim that for the population of heights for presidents and their main​ opponents, the differences have a mean greater than 0 cm.   In this​ example, μd is the mean value of the differences d for the population of all pairs of​ data, where each individual difference d is defined as the​ president's height minus their main​ opponent's height. What are the null and alternative hypotheses for the hypothesis​ test?   H0​: μd equals= greater than> less than< not equals≠ _________ CM       H1​: μd less than< equals= greater than> not equals≠ ________ cm ​(Type integers or decimals. Do not​ round.)     Identify the test statistic.   t=_____ ​(Round to two decimal places as​ needed.)     Identify the​ P-value.   ​P-value=_________ ​(Round to three decimal places as​ needed.)   What is the conclusion based on the hypothesis​ test?   Since the​ P-value is greater than less than or equal to the significance​ level, ▼   fail to reject reject the null hypothesis. There ▼   is is not sufficient evidence to support the claim that presidents tend to be taller than their opponents.

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A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights​ (in centimeters) of randomly selected presidents along with the heights of their main opponents. Complete parts​ (a) and​ (b) below.
 
Height (cm) of President
178
183
182
180
200
179
 
Height (cm) of Main Opponent
170
189
179
180
195
172
 
a. Use the sample data with a
0.01
significance level to test the claim that for the population of heights for presidents and their main​ opponents, the differences have a mean greater than 0 cm.
 
In this​ example,
μd
is the mean value of the differences d for the population of all pairs of​ data, where each individual difference d is defined as the​ president's height minus their main​ opponent's height. What are the null and alternative hypotheses for the hypothesis​ test?
 
H0​:
μd
equals=
greater than>
less than<
not equals≠
_________ CM
 
 
 
H1​:
μd
less than<
equals=
greater than>
not equals≠
________ cm
​(Type integers or decimals. Do not​ round.)
 
 
Identify the test statistic.
 
t=_____
​(Round to two decimal places as​ needed.)
 
 
Identify the​ P-value.
 
​P-value=_________
​(Round to three decimal places as​ needed.)
 
What is the conclusion based on the hypothesis​ test?
 
Since the​ P-value is
greater than
less than or equal to
the significance​ level,
 
fail to reject
reject
the null hypothesis. There
 
is
is not
sufficient evidence to support the claim that presidents tend to be taller than their opponents.
 
 
b. Construct the confidence interval that could be used for the hypothesis test described in part​ (a). What feature of the confidence interval leads to the same conclusion reached in part​ (a)?
 
The confidence interval is
_______cm<μd<_______cm.
​(Round to one decimal place as​ needed.)
 
What feature of the confidence interval leads to the same conclusion reached in part​ (a)?
 
Since the confidence interval contains
 
only negative numbers,
only positive numbers,
zero,
 
fail to reject
reject
the null hypothesis.
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