91Ó°ÊÓ

We need to find which test to carry out either a two-sample t test comparing Try 1 with Try 2 or a paired t test using Gain.

The researcher wants to know if the students are coached and if so, whether their scores will improve.

To use the paired t test if every value in one sample has a corresponding value in the other sample; otherwise, use the two-sample t test.
As each subject is dependent, we will use the paired t test in this scenario.

We need to draw conclusion from test.

As given,

$$\begin{gathered} \mathrm{Î\pm }=0.05 \\ n=427 \\ \overline{x_D}=29 \\ {s}_D=59 \end{gathered}$$

Now, all conditions Random, Independent and Normal are fulfilled then it is right to take hypothesis test.

Mean difference is positive. So, it is either null hypothesis or alternative hypothesis.

$$\begin{gathered} H_0:{\mathrm{μ}}_D=0 \\ {H}_a:{\mathrm{μ}}_D>0 \end{gathered}$$

Statistics of test :

$$\begin{gathered} t=\frac{\overline{x_D}-{\mathrm{μ}}_D}{{\displaystyle \frac{s_D}{\sqrt{n}}}} \\ =\frac{29-0}{{\displaystyle \frac{59}{\sqrt{427}}}} \\ =10.157 \end{gathered}$$

Now,

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

Check $df=100$, P-value will be :

$P<0.0005$

P-value is less than significance level then reject null hypothesis,

Then $\mathrm{P}<0.05⇒\mathrm{Reject}{\mathrm{H}}_0$

Therefore, there is convincing evidence that students who are coached increased their scores significantly.