i want to check if there is a significant difference between in gasoline consumption between gas-1 and gas-2:
here are some observations from gas-1 $\{0.87,0.93,0.82,0.95,0.86,0.88,0.84,0.91,0.86,0.92,0.89,0.90 \}$
and here are some observations from gas-2 $\{0.88,0.96,0.92,0.94,0.93,0.90,0.95,0.92,0.89,0.94,0.97 \}$
i have ordered these as follows:
$\{0.82,0.84,0.86,0.86,0.87,0.88,0.88,0.89,0.89,0.90,0.90,0.91,0.92,0.92,0.92,0.93, 0.93,0.94,0.94,0.95,0.95,0.96,0.97 \}$
the rank sum for gas-1:= $103.5 $ and for gas-2:= $172.5 $
i want to check the hypothesis $H_0:G_1 = G_2$ against $H_1:$ $G_1 \neq G_2$
I know that for a large number of observations i can use: $G_2 \approx N\big( \frac{m_2(N+1)}{2},\frac{m_2n(N+1)}{12} \big)$ where $m_2$ denotes the cardinal number of observations that belong to $G_2$ and $N$ is the total amount of observations. similarly we have the approximation for $G_1 \approx N\big( \frac{m_1(N+1)}{2},\frac{m_1n(N+1)}{12} \big)$.
my questions is if there is a better way than just using the normal approximation here as i proceed to calculate the P-value, in case of using the normal approximation i would get $Z = G_2 - G_1$ and from our observations we got that the difference is 69 so i would have to calculate $\vert Z \vert \geq 69$ (using the differens between two normal approximations)
The test statistic for the Mann-Whitney U test is given by $U = \min(U_1,U_2)$, where $U_1 = G_1 - \frac{n_1(n_1+1)}{2} = 103.5 - 78 = 25.5$ and $U_2 = G_2 - \frac{n_2(n_2+1)}{2} = 172.5 - 66 = 106.5$. We find $U = 25.5$. Consulting a table with the critical values for the test, we find that the critical value for $(n_1,n_2) = (12,13)$ equals $41$ for $\alpha = 0.05$. Since $25.5 < 41$ we reject the hypothesis that there is a difference between the median consumption of gas-1 and gas-2 ($p>0.05$).
Note that using a statistical table for the Mann-Whitney-U test is more exact than using a normal approximation. The downside is that we cannot determine the exact $p$-value.