Answered: Find the test statistic and P-value.…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

Many runners believe that listening to music while running enhances their performance. The authors of a paper wondered if this is true for experienced runners. They recorded time to exhaustion for 11 triathletes while running on a treadmill at a speed determined to be near their peak running velocity. The time to exhaustion was recorded for each participant on two different days. On one day, each participant ran while listening to music that the runner selected as motivational. On a different day, each participant ran with no music playing.

You can assume that it is reasonable to regard these 11 triathletes as representative of the population of experienced triathletes. Only summary quantities were given in the paper, but the data in the table below are consistent with the means and standard deviations given in the paper.

Runner	Time to exhaustion (in seconds)
Motivational music	No music
1	535	468
2	533	449
3	528	481
4	524	573
5	432	561
6	497	591
7	555	473
8	399	499
9	536	551
10	541	500
11	523	524

Do the data provide convincing evidence that the mean time to exhaustion for experienced triathletes is greater when they run while listening to motivational music? Test the relevant hypotheses using a significance level of 0.05. (Use

?_d = ?_music − ?_{no music}.)

State the appropriate null and alternative hypotheses.

H₀: ?_d = 0

H_a: ?_d ≠ 0

H₀: ?_d > 0

H_a: ?_d < 0

H₀: ?_d = 0

H_a: ?_d < 0

H₀: ?_d < 0

H_a: ?_d > 0

H₀: ?_d = 0

H_a: ?_d > 0

Find the test statistic and P-value. (Round your test statistic to one decimal place and your P-value to three decimal places.)

t=P-value=

State the conclusion in the problem context.

We reject H₀. The data do not provide convincing evidence that the mean time to exhaustion for experienced triathletes is greater when they run while listening to motivational music.We reject H₀. The data provide convincing evidence that the mean time to exhaustion for experienced triathletes is greater when they run while listening to motivational music. We fail to reject H₀. The data do not provide convincing evidence that the mean time to exhaustion for experienced triathletes is greater when they run while listening to motivational music.We fail to reject H₀. The data provide convincing evidence that the mean time to exhaustion for experienced triathletes is greater when they run while listening to motivational music.

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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