Question

In a manual on how to have a number one song, it is stated that a song must be no longer than 210 seconds. A simple random sample of 40 current hit songs results in a mean length of 241.4 sec. and a standard deviation of 57.59 sec. Use a 0.05 significance level and the accompanying minitab display to test the claim that the sample is from a population of songs with a mean great thatn 210 sec. What do these results suggest about the advice given in the manual.

The mini tab displays the following:

One-sample T

Test of mu=210 vs.>210

N Mean St. Dev SE Mean 95% lower bound T p

40 241.40 57.59 9.11 226.06 3.45 0.001

A H0 u>210 sec. H1 u < 210sec

B H0 u=210 sec. H1 u < 210sec

C H0 u<210 sec. H1 u> 210sec

D H0 u=210 sec. H1 u> 210sec

Identify the test statistic:

T =

Identify the P-Value

P-value=

Stat the final conclusion that addresses the original claim. Choose from below:

A. Reject H0. There is insufficient evidence to support the claim that the sample is from a population of songs with a mean length greater than 210 sec.
B. Fail to reject H0. There is insufficient evidence to support the claim that the sample is from a population of songs with a mean length great thatn 210 sec.
C. Reject H0. There is sufficient evidence to spport the claim that the sample is from a population of songs with a mean length greater than 210 sec.
D. Fail to reject H0. There is sufficient evidence to support the claim tha tthe sample is from a population of songs with a mean lenght greater than 210 sec.

What do the results suggest about the advice given in the manual?

A. The results do not suggest that the advice of writing a song that must be no longer than 210 seconds is not sound advice.
B The results suggest that the advice of writing a song that must be no longer than 210 seconds is not sound advice
C. The results suggest that 241.4 seconds is the best song lenght.
D. The results are inconclusive because the average length of a hit song is constantly changing.

Answer 1

Answer:

D H0 u=210 sec. H1 u> 210sec

$t=\frac{241.4-210}{\frac{57.59}{\sqrt{40}}}=3.448$    

$p_v =P(t_{(39)}>3.448)=0.000684$  

C. Reject H0. There is sufficient evidence to spport the claim that the sample is from a population of songs with a mean length greater than 210 sec.

Step-by-step explanation:

Data given and notation  

$\bar X=241.4$ represent the sample mean

$s=57.59$ represent the sample standard deviation for the sample  

$n=40$ sample size  

$\mu_o =210$ represent the value that we want to test

$\alpha=0.05$ represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

$p_v$ represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is greater than 210 seconds, the system of hypothesis would be:  

Null hypothesis:$\mu \leq 210$  

Alternative hypothesis:$\mu > 210$  

If we analyze the size for the sample is > 30 but we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$  (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic

We can replace in formula (1) the info given like this:  

$t=\frac{241.4-210}{\frac{57.59}{\sqrt{40}}}=3.448$    

P-value

The first step is calculate the degrees of freedom, on this case:  

$df=n-1=40-1=39$  

Since is a one side test the p value would be:  

$p_v =P(t_{(39)}>3.448)=0.000684$  

Conclusion  

If we compare the p value and the significance level given $\alpha=0.05$ we see that $p_v<\alpha$ so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the mean is significantly higher than 210 seconds.  

C. Reject H0. There is sufficient evidence to spport the claim that the sample is from a population of songs with a mean length greater than 210 sec.