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 President176174182184183167 Height (cm) of Main Opponent167187175180183178 a. Use the sample data with a0.05significance 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,μdis 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 not equals≠less than<equals=greater than>nothingcmH1:μd greater than>equals=not equals≠less than<nothingcm Identify the test statistic. t=Identify the P-value. P-value=What is the conclusion based on the hypothesis test? Since the P-value is less than or equal togreater thanthe significance level, rejectfail to rejectthe null hypothesis. There is notissufficient 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 isnothingcm<μd<cm.What feature of the confidence interval leads to the same conclusion reached in part (a)? Since the confidence interval contains zero,only negative numbers,only positive numbers, fail to rejectrejectthe null hypothesis.

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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	176	174	182	184	183	167
Height (cm) of Main Opponent	167	187	175	180	183	178

a. Use the sample data with a

0.05

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

not equals≠

less than<

equals=

greater than>

nothing

H1:

μd

greater than>

equals=

not equals≠

less than<

nothing

Identify the test statistic.

Identify the P-value.

P-value=

What is the conclusion based on the hypothesis test?

Since the P-value is

less than or equal to

greater than

the significance level,

reject

fail to reject

the null hypothesis. There

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

nothing

cm<μd<

cm.

What feature of the confidence interval leads to the same conclusion reached in part (a)?

Since the confidence interval contains

zero,

only negative numbers,

only positive numbers,

fail to reject

reject

the null hypothesis.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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What feature of the confidence interval leads to the same conclusion reached in part​ (a)? Since the confidence interval contains zero, only negative numbers, only positive numbers, fail to reject reject the null hypothesis

What feature of the confidence interval leads to the same conclusion reached in part (a)? Since the confidence interval contains zero, only negative numbers, only positive numbers, fail to reject reject the null hypothesis