Exercise :
Five random concentration measurements have been taken from each of two points $A$ and $B$ of a river to check the pollution levels.
Point $A$ : $5.2 , 4.8, 5.0, 4.9, 5.1$
Point $B$ : $4.7 , 5.0, 4.9, 4.8, 4.9$
(a) Are there serious evidence at the $1\%$ significance that the mean concentration at Point $B$ is less than the mean concentration at the Point $A$ ? Consider equal distributions for the concentrations at both Points of the river. What other hypothesis is/are needed ?
(b) Find the p-value of the test.
(c) Find a confidence interval with trust 95% for the standard deviation of the concentration at Point $A$.
Attempt :
Let $μ_1$ : be the mean concentration of Point $A$
Let $μ_2$ : be the mean concentration of Point $B$
The statistical hypothesis testing will be :
$H_0 : μ_1 - μ_2 = 0$ against $H_1 : μ_1 - μ_2 > 0$
The number of the degrees of freedom is : $n_1 + n_2 - 2 = 5 + 5 - 2 = 8$
Now :
$$D_1 = 0.5, D_2 = -0.2, D_3 = 0.1, D_4 = 0.1, D_5 = 0.2$$
So: $$μ(D)=0.14$$
And :
$$S_D^2 = 0.063$$
So :
$$t= \frac{μ(D) - δ_ο}{S_D/\sqrt{n}}=\frac{0.14-0}{\sqrt{0.063}/\sqrt{5}}= 1.24722$$
I will have to check :
$$t \geq t_{a;n-1} \Rightarrow 1.24722 \geq t_{0.1;4}$$
and it will lead me to if it's true or not.
Now, I would like to ask $(1)$ is my approach correct ? And $(2)$ why does it ask if any more hypothesis are needed ?
For (b) and (c) I completely have no clue, so I would really appreciate a thorough solution and explanation, because it is one of the type of exercises I have to work for my end-term exams tomorrow.
Thanks in advance for your time ! I would really appreciate some help !
Hint: Read up on two-sample pooled T-test then go through the same set of steps I went through in my previous answer, but using the null distribution for the pooled t test provided in the link. I'm not reproducing the pooled t-test link content here because its (a) good as is and (b) long.