Testing A Hypothesis

[Agent Smith]

4 Spanish teachers take an intensive summer course in Spanish. Their pre-course and post-course scores in Spanish are given below:

Subject	Pre-course score	Post-course score
1	30	29
2	26	30
3	30	35
4	15	18

The course organizers hope to show that taking their course improves Spanish for the subjects.
Formulate a null hypothesis and an alternative hypothesis and test them.
Significance level $\alpha = 0.10$

Construct a 90% confidence interval for the mean increase in scores.

Computations I did:
Mean pre-course score = $\mu_1 = 25.25$
Standard deviation for pre-course score = $\sigma_1 = 6.14$

Mean post-course score = $\mu_2 = 28$
Standard deviation for post-course score = $\sigma_2 = 6.2$

What I did:
For $\mu_2 = 28$, z score = $\frac{28 - 25.25}{6.14}$. Now that I think of it, this is wrong and so I won't bother posting the rest of what I did wrong.

What I should've done:
$\mu_2 - \mu_1 = \mu_D$, the mean difference between pre-course and post-course scores
$\mu_D = 2.75$
$\sigma_D ^2 = \sigma_1 ^2 + \sigma_2 ^2 = 6.14^2 + 6.2^2 \implies \sigma_D = 8.73$

Now to formulate the hypotheses:
Assumption: If there's no difference the post-course distribution of scores should be identical to the pre-course distribution of scores and so:
$H_0: \mu_D = \mu_1 - \mu_1 = 0$.
The standard deviation for the difference of means = $\sigma_1$.

$H_a: \mu_D > 0$

z score for a difference in mean scores of $2.75$ is given by $\frac{2.75 - 0}{6.14} = 0.45$
P-value associated with z score = $0.45$ is $1 - 0.6736 = 0.3264$
P-value > $\alpha$ and so we can't reject $H_0$.

I'm not sure I got this right.


???

Constructing the 90% confidence interval for the mean increase in scores:
For a 90% confidence interval (CI), we need the z-score for 0.95 probability, which is 1.64
So the 90% CI is $\mu_D \pm z^* \times \sigma_D = 2.75 \pm (1.64 \times 8.73) = 2.75 \pm 14.32$
The 90% CI is $[-11.57, 17.07]$

The computed 90% CI seems to square with my conclusion that I can't reject $H_0$ as $0$ lies between $-11.57$ and $17.07$. It is plausible that there's no difference between pre-course and post-course test. In addition, there's even a chance that the course impacts negatively on Spanish language skills ($-11.57 < 0$

Sorry for the long post, but I really need help on this. Is this correct?

 

[mario99]

Are the two samples, $\text{Pre and Post}$, paired or independent?

I see them paired. What about you?

 

[Agent Smith]

I don't know.