Question

Conventional wisdom suggests that Progressive Rock , or "Prog-Rock" - bands that do not regularly get their songs played on the radio - write and record songs that are longer than other "Hard Rock" bands that receive regular radio airplay. The reason being that "prog-rock" bands songs are played less often than "hardrock" bands songs, since song-length is not limited to fit in radio-station format and allow for commercials between air-play.

To test this claim, a statistics professor randomly selected songs from two of his playlists on his iPod, the "ProgRock" playlist and the "HardRock" playlist, and recorded the length of each song in minutes.
The data is found in the accompanying data file. Use α=0.05α=0.05 for all calculations.

Prog_Rock_Sample	Hard_Rock_Sample
4.544	4.386
5.599	5.04
4.333	4.644
3.939	4.211
4.718	3.631
2.868	4.373
4.549	4.133
4.489	4.727
5.292	4.368
4.831	3.759
4.502	3.378
5.306	4.059
5.879	3.75
5.532	3.796
3.487	3.602
2.759	4.712
3.923	3.726
5.517	5.151
4.516	4.103
4.724	4.433
4.295	5.152
5.372	3.77
5.687	5.594
4.713	4.313
4.168	3.549
6.537	5.8
4.938	3.597
	3.717
	3.681
	4.289

Let μPRμPR represent the mean length of songs in the professor's "ProgRock" playlist, and μHRμHR be the mean length of songs in his "HardRock" playlist.

(a) Construct the most appropriate statistical hypotheses that will test this professor's claim.
A. H0:μPR−μHR≥0HA:μPR−μHR<0H0:μPR−μHR≥0HA:μPR−μHR<0
B. H0:μPR−μHR>0HA:μPR−μHR<0H0:μPR−μHR>0HA:μPR−μHR<0
C. H0:μPR−μHR≥0HA:μPR−μHR>0H0:μPR−μHR≥0HA:μPR−μHR>0
D. H0:μPR−μHR=0HA:μPR−μHR<0H0:μPR−μHR=0HA:μPR−μHR<0
E. H0:μPR−μHR=0HA:μPR−μHR>0H0:μPR−μHR=0HA:μPR−μHR>0


(b) Using technology available to you, does there appear to be a difference in the variation in song-length between progressive rock bands and heavy rock bands?
A. There appears to be less variation in the song-length of progressive rock bands when compared to heavy rock bands.
B. There appears to be equal variation in the song-length when comparing progressive rock bands and heavy rock bands.
C. There appears to be more variation in the song-length of progressive rock bands when compared to heavy rock bands.
D. There appears to be unequal variation in the song-length when comparing progressive rock bands to heavy rock bands.


(c) Report the p-value of the test you ran in (b) use at least three decimals in your answer.
P-value =

(d) Test the statistical hypotheses in (a) by carrying out the appropriate statistical test. Find the value of the test statistic for this test, use at least two decimals in your answer.
Test Statistic =

(e) Determine the PP-value of your statistical test in part (d), and report it to at least three decimal places.
P=

(f) Determine the appropriate degrees of freedom for the test in part (d). As an integer.
DF=

(g) Using an αα of 55%, this data suggest that the null hypothesis should  ? be rejected. not be rejected.  Progressive rock songs are  ? on average, longer on average, not longer  when compared to the length of heavy rock songs.

Answer 1

BY USING SPSS WE HAVE

SOLUTION A: NULL HYPOTHESIS H0:μPR−μHR≥0

ALTERNATIVE HYPOTHESIS HA:μPR−μHR>

SOLUTION B: FROM ABOVE TABLE SEE LEVEN'S TEST. P VALUE= 0.235 >0.05 THEREFORE VARIATIONS ARE EQUAL.

There appears to be equal variation in the song-length when comparing progressive rock bands and heavy rock bands. OPTION B

SOLUTION C: P VALUE= 0.235 ( From leven's test)

SOLUTION D: The value of test statistic t is 2.284

e) P value for one tailed test= 0.026/2= 0.013

P value= 0.013

f) Degrees of freedom= n1+n2-2= 27+30-2=55

g) Since P value= 0.013 Smaller than the 0.05 level of significance therefore SIGNIFICANT.

Decision: REJECT NULL HYPOTHESIS H0.

Conclusion:  Progressive rock songs are on average, longer when compared to the length of heavy rock songs.