A music industry producer wondered whether there is a differ

A music industry producer wondered whether there is a difference in lengths (in seconds) of rhythm & blues songs and rap songs. He obtained a random sample of 30 rhythm & blues songs and documented the song lengths. He found that the mean length time (in seconds) of this sample was 242.7 seconds with a standard deviation 26.8. He also took a random sample of 36 rap songs and found that the mean length time of this sample was 237.5 seconds with a standard deviation 28.5.

You will perform a hypothesis test at 0.05 significance level to test the claim that the length of rhythm & blues songs is different from the length of rap songs.

a) State the null and alternate hypotheses using the correct symbols.

b) State the type of test to be used and the requirements needed for the test.   

c) Find the test statistic.

d) Find the critical value OR the p-value of the test.

e) State a statistical decision and interpret your decision in the context of the problem.

f) Construct a 95% confidence interval for the difference between the mean time length of rhythm & blues songs and the mean time length of rap songs. Show your work or calculator command.

g) Explain how your conclusions from the confidence interval and the hypothesis test are the same or different.

Solution

Set Up Hypothesis
Null , There Is No-Significance between them Ho: u1 = u2
Alternate , There Is Significance between them - H1: u1 != u2
Test Statistic
X(Mean)=242.7
Standard Deviation(s.d1)=26.8 ; Number(n1)=30
Y(Mean)=237.5
Standard Deviation(s.d2)=28.5; Number(n2)=36
we use Test Statistic (t) = (X-Y)/Sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =242.7-237.5/Sqrt((718.24/30)+(812.25/36))
to =0.763
| to | =0.763
Critical Value
The Value of |t | with Min (n1-1, n2-1) i.e 29 d.f is 2.045
We got |to| = 0.76253 & | t | = 2.045
Make Decision
Hence Value of |to | < | t | and Here we Do not Reject Ho
P-Value: Two Tailed ( double the one tail ) - Ha : ( P != 0.7625 ) = 0.452
Hence Value of P0.05 < 0.452,Here We Do not Reject Ho

[ANSWERS]
a.Ho: u1 = u2 , H1: u1 != u2
b.t Test of Difference Means for Unequal Variance
c. to =0.763
d. Crtical value: 2.045, Ha : ( P != 0.7625 ) = 0.452
e. the length of rhythm & blues songs is n\'t different from the length of rap songs.