To determine the estimated standard error for the given repeated-measures study, follow these steps:
1. Identify the given information:
- Number of participants, [tex]\( n = 4 \)[/tex]
- Sum of squares for the difference scores, [tex]\( SS = 75 \)[/tex]
- Mean of the difference scores, [tex]\( M_D = 2 \)[/tex] (While this value is given, it's not directly used in the calculation of standard error.)
2. Calculate the variance of the difference scores:
The formula for variance ([tex]\( s^2_D \)[/tex]) using the sum of squares (SS) and the number of participants (n) is:
[tex]\[
s^2_D = \frac{SS}{n - 1}
\][/tex]
Plugging in the known values:
[tex]\[
s^2_D = \frac{75}{4 - 1} = \frac{75}{3} = 25
\][/tex]
3. Calculate the estimated standard error:
The estimated standard error ([tex]\( s_{M_D} \)[/tex]) is obtained by dividing the variance by the number of participants and then taking the square root of the result:
[tex]\[
s_{M_D} = \sqrt{\frac{s^2_D}{n}}
\][/tex]
Using the variance calculated in the previous step:
[tex]\[
s_{M_D} = \sqrt{\frac{25}{4}} = \sqrt{6.25} = 2.5
\][/tex]
Thus, the estimated standard error for this repeated-measures study is [tex]$2.5$[/tex].
