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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Description: 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)

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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 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.

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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