To solve this question, we need to calculate the estimated standard error [tex]\( S_{M_D} \)[/tex] and the t value step-by-step. Let's start with the information given:
- Sample size, [tex]\( n = 9 \)[/tex]
- Mean of the difference scores, [tex]\( M_D = 4.9 \)[/tex]
- Sum of squares of the difference scores, [tex]\( SS = 72 \)[/tex]
1. Calculate the variance of the difference scores:
[tex]\[
\text{Variance}_D = \frac{SS}{n - 1}
\][/tex]
Here, [tex]\( n - 1 = 9 - 1 = 8 \)[/tex],
[tex]\[
\text{Variance}_D = \frac{72}{8} = 9
\][/tex]
2. Calculate the standard deviation of the difference scores:
[tex]\[
\text{Standard Deviation}_D = \sqrt{\text{Variance}_D}
\][/tex]
[tex]\[
\text{Standard Deviation}_D = \sqrt{9} = 3
\][/tex]
3. Calculate the estimated standard error [tex]\( S_{M_D} \)[/tex]:
[tex]\[
S_{M_D} = \frac{\text{Standard Deviation}_D}{\sqrt{n}}
\][/tex]
Here, [tex]\( \sqrt{n} = \sqrt{9} = 3 \)[/tex],
[tex]\[
S_{M_D} = \frac{3}{3} = 1
\][/tex]
4. Calculate the t value:
[tex]\[
t = \frac{M_D}{S_{M_D}}
\][/tex]
[tex]\[
t = \frac{4.9}{1} = 4.9
\][/tex]
Therefore, the estimated standard error [tex]\( S_{M_D} \)[/tex] is 1 and the t value is 4.9. So the correct answers are:
[tex]\[ S_{M_D} = 1 \][/tex] and [tex]\[ t = 4.9 \][/tex]
