Chapter 10.3, Problem 101E

(a)

To determine

To find out which is the correct test and why.

Answer to Problem 101E

Paired t test will be used.

Explanation of Solution

The researcher wants to find out if the students are coached then their scores will be higher or not.

So if every value in one sample has a corresponding value in the other sample then we use the paired t test else we use the two-sample t test.

Thus, in this case we will be using the paired t test as each subjects are dependent.

(b)

To determine

To carry out the proper test and what do you conclude.

Answer to Problem 101E

There is a convincing evidence that the cores of students who are coached increase significantly.

Explanation of Solution

It is given:

  $\begin{array}{l}\alpha =0.05\\ n=427\\ {\overline{x}}_D=29\\ s_D=59\end{array}$

Since all the three conditions: Random, Independent and Normal conditions are satisfied then it is appropriate to conduct the hypothesis test.

The given claim is: mean difference is positive. So, the claim is either null hypothesis or alternative hypothesis. That is,

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

The test statistics value will be:

  $\begin{array}{l}t=\frac{{\overline{x}}_D-{\mu }_D}{\frac{s_D}{\sqrt{n}}}\\ =\frac{29-0}{\frac{59}{\sqrt{427}}}\\ =10.157\end{array}$

For the P -value he degree of freedom will be as:

  $\begin{array}{l}df=n-1\\ =427-1\\ =426\end{array}$

We will check the $df=100$ instead. Hus, the P -value will be:

  $P<0.0005$

If the P -values is less than the significance level we reject the null hypothesis, then we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there is convincing evidence that the cores of students who are coached increase significantly.