Chapter 10.3, Problem 87E

(a)

To determine

To explain why or why not can you conclude that the difference in the ability to memorize words was caused by whether students were performing the task in silence or with playing music.

Answer to Problem 87E

Yes.

Explanation of Solution

As we know that the selected students are a random sample while the order of the treatments were randomly assigned to the students.

This then implies that the experiment is a randomized comparative experiment and then causation is proven when we find a statistically significant difference because the result need to be due to the treatments itself as the randomization ensures that the difference cannot be due to the other variables beside the treatment.

(b)

To determine

To explain do these data provide convincing evidence at the $\alpha =0.01$ level that the number of words recalled in silence or when listening to music differs on average for students at the school.

Answer to Problem 87E

There is a convincing evidence at the $\alpha =0.01$ level that the number of words recalled in silence or when listening to music differs on average for students at the school.

Explanation of Solution

It is given that:

  $\begin{array}{l}\alpha =0.01\\ n=30\\ {\overline{x}}_D=1.57\\ s_D=2.70\end{array}$

Given claim is that: Difference.

The claim is either null hypothesis or an alternative hypothesis.

  $\begin{array}{l}{H}_0:{\mu }_D=0\\ H_a:{\mu }_D\ne 0\end{array}$

The test statistics value will be:

  $\begin{array}{l}t=\frac{{\overline{x}}_D-{\mu }_D}{\frac{s_D}{\sqrt{n}}}\\ =\frac{1.57-0}{\frac{2.70}{\sqrt{30}}}\\ =3.185\end{array}$

And the degree of freedom will be:

  $\begin{array}{l}df=n-1\\ =30-1\\ =29\end{array}$

As we know that the P -value is the probability of obtaining the sample results or results more extreme when the null hypothesis is true.

  $0.002=2(0.001)<P<2(0.0025)=0.005$

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

  $P<0.01\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there is a convincing evidence at the $\alpha =0.01$ level that the number of words recalled in silence or when listening to music differs on average for students at the school.

(c)

To determine

To explain what type of error could you have made- Type I error or Type II error, based on the conclusion in part (a).

Answer to Problem 87E

Type I error.

Explanation of Solution

We conclude in part (a) and part (b) that,

There is convincing evidence to support the claim that that the number of words recalled in silence or when listening to music differs on average for students at the school.

A type I error occurs if we reject a null hypothesis when the null hypothesis is true. And the Type II error occurs if we fails to reject the null hypothesis when the null hypothesis is false.

Thus, in this case we reject the null hypothesis then it is a Type I error.