Agent Smith
Full Member
- Joined
- Oct 18, 2023
- Messages
- 335
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:
[imath]\text{z score}_\text{Bobby} = \frac{\text{score - mean}}{\text{standard deviation}} = \frac{75 - 80}{5} = -1[/imath]
[imath]\text{z score}_\text{Cindy} = \frac{70 - 75}{10} = -0.5[/imath]
[imath]\text{z score}_\text{Cindy} > \text{z score}_\text{Bobby}[/imath]. 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
[imath]\mu_{X - Y} = \mu_X - \mu_Y = 80 - 75 = 5[/imath]
[imath]\sigma_{X - Y} = \sqrt{5^2 + 10^2} = \sqrt {125} = 5\sqrt 5[/imath]
Bobby's score - Cindy's score = 75 - 70 = 5
[imath]\text{z score}_\text{Bobby - Cindy} = \frac{5 - 5}{5\sqrt 5} = 0[/imath]
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?
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:
[imath]\text{z score}_\text{Bobby} = \frac{\text{score - mean}}{\text{standard deviation}} = \frac{75 - 80}{5} = -1[/imath]
[imath]\text{z score}_\text{Cindy} = \frac{70 - 75}{10} = -0.5[/imath]
[imath]\text{z score}_\text{Cindy} > \text{z score}_\text{Bobby}[/imath]. 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
[imath]\mu_{X - Y} = \mu_X - \mu_Y = 80 - 75 = 5[/imath]
[imath]\sigma_{X - Y} = \sqrt{5^2 + 10^2} = \sqrt {125} = 5\sqrt 5[/imath]
Bobby's score - Cindy's score = 75 - 70 = 5
[imath]\text{z score}_\text{Bobby - Cindy} = \frac{5 - 5}{5\sqrt 5} = 0[/imath]
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?
