91Ó°ÊÓ

It is given:

$$\begin{gathered} \stackrel{¯}{x}=3.618 \\ n=36 \\ s=3.055 \\ c=0.90 \end{gathered}$$

The degree of freedom will be:

$$\begin{gathered} df=n-1 \\ =36-1 \\ =35 \end{gathered}$$

Now, the value of t will be:

$t_{\mathrm{Î\pm }/2}=1.697$

Now, the confidence interval will be:

$$\begin{gathered} \stackrel{¯}{x}-t_{\mathrm{Î\pm }/2}×\frac{s}{\sqrt{n}} \\ =3.618-1.697×\frac{3.055}{\sqrt{36}} \\ =2.7539 \\ \stackrel{¯}{x}+t_{\mathrm{Î\pm }/2}×\frac{s}{\sqrt{n}} \\ =3.618+1.697×\frac{3.055}{\sqrt{36}} \\ =4.4821 \end{gathered}$$

Thus we conclude that we are$95\%$ confident that the true mean reasoning score after six months of piano lessons is between$2.7539and4.4821$ higher than the true mean reasoning before six months of piano lessons.

A completely randomized experiment randomly assigns all subjects to a group.

In this case, the experiment is not completely randomized since we use the subjects before and after the treatment. Thus the experiment is not a completely randomized experiment.

We cannot prove causation if we do not use a completely randomized experiment because it is possible that the causation of the difference is another variable.

For example, age could be the cause because a child will most likely have a better reasoning skills after six months.