Chapter 24, Problem 37E

(a)

To determine

To test an appropriate hypothesis and state your conclusion.

Answer to Problem 37E

It does not matter one listen to rap music while studying or not.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=10\\ {\overline{x}}_2=12.77\\ n_1=20\\ n_2=13\\ s_1=3.19\\ s_2=4.73\end{array}$

Now, let us define the hypotheses for testing:

  $\begin{array}{l}{H}_0:{\mu }_1={\mu }_2\\ H_1:{\mu }_1<{\mu }_2\end{array}$

Thus, the value of test statistics is:

  $\begin{array}{l}t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\\ =\frac{10-12.77}{\sqrt{\frac{{3.99}^2}{29}+\frac{{4.73}^2}{13}}}\\ =-1.36\end{array}$

And the value of degree of freedom is:

  $\begin{array}{l}df=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})}{\frac{{(\raisebox{1ex}{$s_1^2$}\!\left/ \!\raisebox{-1ex}{$n_1$}\right.)}^2}{n_1-1}+\frac{{(\raisebox{1ex}{$s_2^2$}\!\left/ \!\raisebox{-1ex}{$n_2$}\right.)}^2}{n_2-1}}\\ =20\end{array}$

Thus, the P-value will be:

  $0.05<P<0.10$

And as we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected. So,

  $P>0.05\Rightarrow \text{Fail to Reject }{H}_0$

Thus, there is not sufficient evidence to support the claim of an improvement.

(b)

To determine

To estimate the size of that difference with a confidence interval and explain what your interval means.

Answer to Problem 37E

It is not possible.

Explanation of Solution

It is given that:

  $\begin{array}{l}{\overline{x}}_1=10\\ {\overline{x}}_2=10.72\\ n_1=20\\ n_2=29\\ s_1=3.19\\ s_2=3.99\end{array}$

  $P>0.05\Rightarrow \text{Fail to Reject }{H}_0$

Thus, there is not sufficient evidence to support the claim of an improvement. So, it is not better to study while listening to Mozart than to rap music. Thus, it is not possible because there was no significant difference in part (a).