Chapter 10.3, Problem 81E

(a)

To determine

To construct and interpret a $90\%$ confidence interval for the true mean difference.

Answer to Problem 81E

We are $90\%$ confident that the true mean reasoning score after six months of piano lessons is between $2.7539\text{ and }4.4821$ higher than the true mean reasoning before six months of piano lessons.

Explanation of Solution

It is given:

  $\begin{array}{l}\overline{x}=3.618\\ n=36\\ s=3.055\\ c=0.90\end{array}$

Now, the degree of freedom will be:

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

Now, the value of t will be:

  $t_{\alpha /2}=1.697$

Now, the confidence interval will be:

  $\begin{array}{l}\overline{x}-t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =3.618-1.697\times \frac{3.055}{\sqrt{36}}\\ =2.7539\\ \overline{x}+t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =3.618+1.697\times \frac{3.055}{\sqrt{36}}\\ =4.4821\end{array}$

Thus we conclude that we are $90\%$ confident that the true mean reasoning score after six months of piano lessons is between $2.7539\text{ and }4.4821$ higher than the true mean reasoning before six months of piano lessons.

(b)

To determine

To explain can you conclude that taking six months of piano lessons would cause an increase in preschool students’ average reasoning scores or not.

Answer to Problem 81E

No, we cannot conclude.

Explanation of Solution

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.