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. For purposes of this exercise, 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 1 535 466 2 533 447 3 526 481 4 524 573 5 432 561 6 498 591 7 555 473 8 397 497 9 539 551 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 α = 0.05. (Use μd = μmusic − μno music.) State the appropriate null and alternative hypotheses. Find the test statistic and P-value. (Use a table or technology. Round your test statistic to two decimal places and your P-value to three decimal places.) t = P-value =

Answer 1

The table given below ,

Motivational Music (X)	No Music(Y)	di=X-Y	di^2
535	466	69	4761
533	447	86	7396
526	481	45	2025
524	573	-49	2401
432	561	-129	16641
498	591	-93	8649
555	473	82	6724
397	497	-100	10000
539	551	-12	144
541	500	41	1681
523	524	-1	1
	Total	-61	60423

From table ,

$s_d=\sqrt{\frac{\sum di^2-n*\bar{d}^2}{n-1}}=\sqrt{\frac{60423-11*(-5.5455)^2}{11-1}}=77.5143$

Hypothesis : $H_0:\mu_d=0$ VS $H_a:\mu_d>0$

The test statistic is ,

$t=\frac{\bar{d}}{s_d/\sqrt{n}}\to t_{n-1}$

$=\frac{-5.5455}{77.5143/\sqrt{11}}=-0.24$

P-value = $P(t_{n-1}>|t_{stat}|)=P(t_{11-1}>0.24)=0.4076$ ; From excel "=TDIST(0.24,10,1)"

Decision : Here , P-value =0.4076 > α = 0.05

Therefore , fail to reject Ho at α = 0.05 significance level.

Conclusion : Hence , the data does not provide convincing evidence that the mean time to exhaustion for experienced triathletes is greater when they run while listening to motivational music