Hi I'm having a statistical question like this.
$A$ and $B$ have taken two different standard English tests. $A$ has scored $1250$ on Test $A$, $N$(mean=$1000$, $SD=200$) and $B$ has scored $15$ on Test $B$, $N$(mean=$11$, $SD=3$). Based on the results which one is better in his English?
I first thought that this can be solved as hypothesis testing. But I don't know the sample size of both tests. Can someone please help me solving this.
Person A scored 1250 in a test with $\mu=1000$ and $\sigma=200$. Thus person A scored 250 points above the mean in test A. Since the standard deviation of test A is $200$, person A scored $\displaystyle\frac{250}{200}=\frac 5 4$ standard deviations above the mean.
Similarly, person B scored $\displaystyle\frac{15-11}{3}=\frac 4 3$ standard deviations above the mean in test B.
Now that the scores are normalized (expressed in the same units, in this case, number of standard deviations above the mean), the results are more easily comparable.
Since person B's score is $\displaystyle\frac 4 3$, which is greater than person A's score of $\displaystyle \frac 5 4$, based on these results, person B received the better score.
If you prefer a formula, the number of standard deviations above the mean (the $z$-score) is:
$$z=\frac{x-\mu}{\sigma}$$