How Do You Calculate Uncertainty in Mean Difference?

[selsunblue]

Homework Statement

Homework Equations

Mean difference and SEM

The Attempt at a Solution

Would the mean difference be 0.16?. I'm not sure about the answer because they give that much working space, Do I have to find the uncertainty of the difference first and then my answer implement that into my answer to be 0.16 +- (uncertainty)?. How would I calculate the uncertainty of the mean difference? Do I just add the SEM's for both means?

 

[mfb]

Would the mean difference be 0.16?

Sure. The large working space is a bit odd.

Do I have to find the uncertainty of the difference first and then my answer implement that into my answer to be 0.16 +- (uncertainty)?

That is part (ii).

How would I calculate the uncertainty of the mean difference? Do I just add the SEM's for both means?

No. Use the method (hopefully) described in section 1.5.

 

[vanceEE]

Do I just add the SEM's for both means?

Here are the basic rules for uncertainty:

Addition/Subtraction
$$(A ± ΔA) + (B ± ΔB) = (A+B) ± (ΔA+ΔB)$$
$$(A ± ΔA) - (B ± ΔB) = (A-B) ± (ΔA+ΔB)$$

When dividing and multiplying, changing to relative uncertainty, then converting back simplifies things.
where: $$ε = \frac{ΔA}{A}*100$$ 'relative uncertainty'

$$(A ± ΔA) / (B ± ΔB) = (A/B) ± (ε_A+ε_B)$$
$$(A ± ΔA)(B ± ΔB) = (AB) ± (ε_A+ε_B)$$

Then you can convert back to abs. uncertainty.

 

[tms]

Here are the basic rules for uncertainty:

Addition/Subtraction
$$(A ± ΔA) + (B ± ΔB) = (A+B) ± (ΔA+ΔB)$$
$$(A ± ΔA) - (B ± ΔB) = (A-B) ± (ΔA+ΔB)$$

Are you sure? I believe the uncertainties should not simply be added, but added in quadrature. That is, if
$$x = au \pm bv,$$
then
$$\sigma_x^2 = a^2 \sigma_u^2 + b^2 \sigma_v^2.$$
This comes from the basic definition: If $x$ is a function of measured variables $u, v, \ldots$,
$$x = f(u, v, \ldots),$$
then,
$$\sigma_x^2 \approx \sigma_u^2 \left ( \frac{\partial x}{\partial u} \right )^2 + \sigma_v^2 \left ( \frac{\partial x}{\partial v} \right )^2 + \ldots \;.$$

I did not check your other equations.

 

[mfb]

Yes, in general the uncertainties should be added in quadrature, if correlations are not important.

A linear addition is sometimes used as worst case estimate to include possible correlations between the uncertainties. We take the difference between measured heights here - there is no way the uncertainties could be correlated "the wrong way" (e. g. a systematic deviation towards larger men and smaller women at the same time).