chosing right confidence interval( left , right , both ) I am learning about confidence interval and hypothesis testing.
Lets demonstrate my problem:
The trader says the car burns 8l per 100 miles of fuel. We have to find out if he does or does not lie.
our $H_{0}$ is $\mu=8$ , and $H_{1}$ is $\mu > 8$
So we took 100 cars and calculated average burnout (couldnt find exact word for this ) per 100 miles its our $Xn$
Now we construct RIGHT confidence  interval, using  $(Xn_{\chi_{\alpha}},\inf)$
Where 
$\chi_{\alpha} = t_{\alpha,n-1} *\frac{standard deviation}{\sqrt(n)}$
We calcaulate it , checks if 8 is in the interval, if yes we accept hypothesis if not we reject if with $\alpha$ probabilty that its wrong.
However i do not understand, why do we chose RIGHT interval and not left? I fail to find logical decision behind it. 
Could someone please explain it to me? If we fin that $Xn$ = 12 , and interval is (9.6 , inf ) We reject hypothesis bcs 8 isnt in the interval. But shoulnt it be good that it does not belong to interval? 
Thanks for answer
 A: You chose "the trader tells the truth" as your null hypothesis $H_0,$
and "the trader lies" as your alternative hypothesis $H_1.$
If you find that $8$ is outside the confidence interval, you will reject the null hypothesis.
Whether this is a "good thing" or a "bad thing" depends on whether you think it is a "good thing" to find out that the trader is lying.
If you chose a left confidence interval instead of a right confidence interval, then your confidence interval would include all values less than $X_n$ and some values greater than $X_n.$
In particular, if $X_n = 12,$ your confidence interval would include $8$ and you would not be able to reject the null hypothesis--you could not say the trader was lying.
Even if every single trial found that the car burned $2000$ liters in $100$ miles during that trial, you still could not reject the null hypothesis,
because $8 < 2000$ and you are using a left confidence interval;
whereas any reasonable person would say it is obvious that the trader lied (and any reasonable statistical analysis would support that statement).
A: A more accurate characterization of this test is "Test if the trader is exaggerating the fuel efficiency". It's assumed you don't care about buying a car with exactly 8 liters per 100 miles fuel efficiency, but only that the car is at least that efficient (you'd be more than happy if it is actually 4 liters per 100 miles). You are correct that a two-tailed test would be the best one to test if the trader is inaccurate (we can't test lying unless you know intent anyway). However, this is not what is actually being tested, so that is a problem with the wording given to you "is trader lying" vs "is trader exaggerating".
The choice of rejection region is not purely mathematical: you need to take into account your losses of being wrong. In this case, if the car burns less than 8 liters per 100 miles (i.e., salesperson was being conservative) then you will simply benefit more than you expected. So, it makes little sense to spend statistical "power" on testing this situation. You want to focus on the outcome that will produce the most losses -- in this case, getting a car that is not as fuel efficient as advertised. 
