Question

Does a statistics course improve a student's mathematics skills,as measured by a national test? Suppose a random sample of 13 students takes the same national mathematics exam prior to enrolling in a stats course and just after completing the course. At a 1% level of significance determine whether the scores after the stats course are significantly higher than the scores before. Take the differences = before - after.

Answer 1

Answer:

There is not enough evidence to support the claim that the scores after the stats course are significantly higher than the scores before (the difference in the scores is higher than 0).

P-value=0.042.

Step-by-step explanation:

The question is incomplete:

The data of the scores for each student is:

Before    After

430        465

485        475

520        535

360        410

440        425

500        505

425        450

470        480

515        520

430        430

450        460

495        500

540        530

We will generate a sample for the difference of scores (before - after) and test that sample.

The sample of the difference is [35 -10 15 50 -15 5 25 10 5 0 10 5 -10]

This sample, of size n=13, has a mean of 9.615 and a standard deviation of 18.423.

The claim is that the scores after the stats course are significantly higher than the scores before (the difference in the scores is higher than 0).

Then, the null and alternative hypothesis are:

$H_0: \mu=0\\\\H_a:\mu> 0$

The significance level is 0.01.

The estimated standard error of the mean is computed using the formula:

$s_M=\dfrac{s}{\sqrt{n}}=\dfrac{18.423}{\sqrt{13}}=5.11$

Then, we can calculate the t-statistic as:

$t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{9.615-0}{5.11}=\dfrac{9.615}{5.11}=1.882$

The degrees of freedom for this sample size are:

$df=n-1=13-1=12$

This test is a right-tailed test, with 12 degrees of freedom and t=1.882, so the P-value for this test is calculated as (using a t-table):

$P-value=P(t>1.882)=0.042$

As the P-value (0.042) is bigger than the significance level (0.01), the effect is  not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that the scores after the stats course are significantly higher than the scores before (the difference in the scores is higher than 0).