How to fairly compare test scores

[Agent Smith]

Bobby scores 75 in class whose average score is 80 with a standard deviation of 5.
Cindy scores 70 in a class whose average score is 75 with a standard deviation of 10.
Who did better in their test?

I've been told to do this:
$\text{z score}_\text{Bobby} = \frac{\text{score - mean}}{\text{standard deviation}} = \frac{75 - 80}{5} = -1$

$\text{z score}_\text{Cindy} = \frac{70 - 75}{10} = -0.5$

$\text{z score}_\text{Cindy} > \text{z score}_\text{Bobby}$. So Cindy did better.

Although Cindy did better compared to Bobby, more than 50% of the girls scored higher than Cindy.
Correct?

What I tried next ...

X = Random variable score in Bobby's class
Y = Random variable score in Cindy's class
$\mu_{X - Y} = \mu_X - \mu_Y = 80 - 75 = 5$

$\sigma_{X - Y} = \sqrt{5^2 + 10^2} = \sqrt {125} = 5\sqrt 5$

Bobby's score - Cindy's score = 75 - 70 = 5

$\text{z score}_\text{Bobby - Cindy} = \frac{5 - 5}{5\sqrt 5} = 0$

Does this mean Bobby's score - Cindy's score = 5 was a typical difference? How else can we interpret this result?
How to combine these two results into a more complete picture of what's actually going on?

 

[Agent Smith]

 

[Agent Smith]

 

[khansaheb]

Who did better in their test?

At what significance level?

 

[Agent Smith]

@khansaheb , I'm not sure. The question doesn't specify such a requirement. How about at 5% significance level? That seems to be the standard for most statistical questions at my level.

 

[Dr.Peterson]

Bobby scores 75 in class whose average score is 80 with a standard deviation of 5.
Cindy scores 70 in a class whose average score is 75 with a standard deviation of 10.
Who did better in their test?

I've been told to do this:
$\text{z score}_\text{Bobby} = \frac{\text{score - mean}}{\text{standard deviation}} = \frac{75 - 80}{5} = -1$

$\text{z score}_\text{Cindy} = \frac{70 - 75}{10} = -0.5$

$\text{z score}_\text{Cindy} > \text{z score}_\text{Bobby}$. So Cindy did better.

Why do you not think this is the appropriate answer ("at your level")? What more are you looking for? (The rest of what you say doesn't seem relevant to the question as asked.)

 

[Agent Smith]

Why do you not think this is the appropriate answer ("at your level")? What more are you looking for? (The rest of what you say doesn't seem relevant to the question as asked.)

I don't know. It seemed natural to compute the $\mu_{X - Y}$, but I was told it's meaningless to do this with the given data. I didn't quite get that. I have a fair idea of what the z-score does it seems.