Does 10K running time decrease when the runner listens to music? Nine runners were timed as they ran a 10K with and without listening to music. The running times in minutes are shown below.

Running Time

With Music 46 47 48 42 44 51 49 46 50

Without Music 50 48 45 43 47 53 49 49 53

Assume a Normal distribution. What can be concluded at the the αα = 0.01 level of significance?

For this study, we should use Select an answer t-test for a population mean z-test for the difference between two population proportions z-test for a population proportion t-test for the difference between two dependent population means t-test for the difference between two independent population means

The null and alternative hypotheses would be:


H0:H0:  Select an answer p1 μ1 μd ? < ≠ > = Select an answer p2 0 μ2 (please enter a decimal)

H1:H1:  Select an answer μd μ1 p1 ? > ≠ = < Select an answer p2 0 μ2 (Please enter a decimal)

The test statistic ? t z = (please show your answer to 3 decimal places.)

The p-value = (Please show your answer to 4 decimal places.)

The p-value is ? > ≤ αα

Based on this, we should Select an answer reject fail to reject accept the null hypothesis.

Thus, the final conclusion is that ...

The results are statistically significant at αα = 0.01, so there is sufficient evidence to conclude that the population mean running time with music is less than the population mean running time without music.

The results are statistically significant at αα = 0.01, so there is sufficient evidence to conclude that the nine runners finished in less time on average with music compared to running without music.

The results are statistically insignificant at αα = 0.01, so there is insufficient evidence to conclude that the population mean running time with music is less than the population mean running time without music.

The results are statistically insignificant at αα = 0.01, so there is statistically significant evidence to conclude that the population mean running time with music is equal to the population mean running time without music.

Interpret the p-value in the context of the study.

If the sample mean running time with music for the 9 runners is the same as the sample mean running time without music for these 9 runners and if another 9 runners are observed running the 10K with and without music then there would be a 2.98% chance of concluding that the mean running time with music for the 9 runners is at least 1.6 minutes less than the mean running time for these 9 runners without music.

If the population mean running time with music is the same as the population mean running time without music and if another 9 runners compete with and without music then there would be a 2.98% chance that the mean running time for the 9 runners would be at least 1.6 minutes less with music compared to them running without music.

There is a 2.98% chance that the mean running time for the 9 runners with music is at least 1.6 minutes less than the mean time for these 9 runners without music.

There is a 2.98% chance of a Type I error.

Interpret the level of significance in the context of the study.

If the population mean running time with music is the same as the population mean running time without music and if another 9 runners compete with and without music then there would be a 1% chance that we would end up falsely concluding that the population mean running time with music is less than the population mean running time without music

There is a 1% chance that the runners aren't in good enough shape to run a 10K, so music is irrelevant.

There is a 1% chance that the population mean running time is the same with and without music.

If the population mean running time with music is the same as the population mean running time without music and if another 9 runners compete in the 10K with and without music, then there would be a 1% chance that we would end up falsely concluding that the sample mean running times with music and without music for these 9 runners differ from each other.

Question

Does 10K running time decrease when the runner listens to music? Nine runners were timed as they ran a 10K with and without listening to music. The running times in minutes are shown below.

Running Time

With Music	46	47	48	42	44	51	49	46	50
Without Music	50	48	45	43	47	53	49	49	53

Assume a Normal distribution. What can be concluded at the the αα = 0.01 level of significance?

For this study, we should use Select an answer t-test for a population mean z-test for the difference between two population proportions z-test for a population proportion t-test for the difference between two dependent population means t-test for the difference between two independent population means

The null and alternative hypotheses would be:

H0:H0: Select an answer p1 μ1 μd ? < ≠ > = Select an answer p2 0 μ2 (please enter a decimal)

H1:H1: Select an answer μd μ1 p1 ? > ≠ = < Select an answer p2 0 μ2 (Please enter a decimal)

The test statistic ? t z = (please show your answer to 3 decimal places.)
The p-value = (Please show your answer to 4 decimal places.)
The p-value is ? > ≤ αα
Based on this, we should Select an answer reject fail to reject accept the null hypothesis.
Thus, the final conclusion is that ...
- The results are statistically significant at αα = 0.01, so there is sufficient evidence to conclude that the population mean running time with music is less than the population mean running time without music.
- The results are statistically significant at αα = 0.01, so there is sufficient evidence to conclude that the nine runners finished in less time on average with music compared to running without music.
- The results are statistically insignificant at αα = 0.01, so there is insufficient evidence to conclude that the population mean running time with music is less than the population mean running time without music.
- The results are statistically insignificant at αα = 0.01, so there is statistically significant evidence to conclude that the population mean running time with music is equal to the population mean running time without music.
Interpret the p-value in the context of the study.
- If the sample mean running time with music for the 9 runners is the same as the sample mean running time without music for these 9 runners and if another 9 runners are observed running the 10K with and without music then there would be a 2.98% chance of concluding that the mean running time with music for the 9 runners is at least 1.6 minutes less than the mean running time for these 9 runners without music.
- If the population mean running time with music is the same as the population mean running time without music and if another 9 runners compete with and without music then there would be a 2.98% chance that the mean running time for the 9 runners would be at least 1.6 minutes less with music compared to them running without music.
- There is a 2.98% chance that the mean running time for the 9 runners with music is at least 1.6 minutes less than the mean time for these 9 runners without music.
- There is a 2.98% chance of a Type I error.
Interpret the level of significance in the context of the study.
- If the population mean running time with music is the same as the population mean running time without music and if another 9 runners compete with and without music then there would be a 1% chance that we would end up falsely concluding that the population mean running time with music is less than the population mean running time without music
- There is a 1% chance that the runners aren't in good enough shape to run a 10K, so music is irrelevant.
- There is a 1% chance that the population mean running time is the same with and without music.
- If the population mean running time with music is the same as the population mean running time without music and if another 9 runners compete in the 10K with and without music, then there would be a 1% chance that we would end up falsely concluding that the sample mean running times with music and without music for these 9 runners differ from each other.

Accepted Answer

From the provided information,
The difference table can be constructed as:

With Music

Without…