To calculate $S_{\text{Difference}}$ for each study, we use the following formula:

$S_{\text{Difference}} = \sqrt{\frac{(N_1 - 1)S_1^2 + (N_2 - 1)S_2^2}{N_1 + N_2 - 2}} \times \sqrt{\frac{1}{N_1} + \frac{1}{N_2}}$

Where:

• $N_1$ and $N_2$ are the sample sizes of two groups,
• $S_1^2$ and $S_2^2$ are the variances of the two groups.

Step-by-step process for Study (a):

• $N_1 = 40$
• $S_1^2 = 2$
• $N_2 = 60$
• $S_2^2 = 5$

1. First, calculate the pooled variance:

$S_{\text{pooled}}^2 = \frac{(N_1 - 1)S_1^2 + (N_2 - 1)S_2^2}{N_1 + N_2 - 2}$

$S_{\text{pooled}}^2 = \frac{(40 - 1) \times 2 + (60 - 1) \times 5}{40 + 60 - 2}$

$S_{\text{pooled}}^2 = \frac{39 \times 2 + 59 \times 5}{98} = \frac{78 + 295}{98} = \frac{373}{98} = 3.806$

2. Now, calculate $S_{\text{Difference}}$:

$S_{\text{Difference}} = \sqrt{3.806} \times \sqrt{\frac{1}{40} + \frac{1}{60}}$

$S_{\text{Difference}} = 1.951 \times \sqrt{0.025 + 0.01667}$

$S_{\text{Difference}} = 1.951 \times \sqrt{0.04167} = 1.951 \times 0.2041 = 0.3983$

Thus, the $S_{\text{Difference}}$ for study (a) is approximately 0.398 (rounded to three decimal places).

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Would you like further details or have any questions?

Here are five related questions to explore further:

1. How would the formula change if the two groups had unequal variances?
2. Can you calculate the effect size for this study based on the given data?
3. What assumptions are necessary for pooling variances in this way?
4. How does the size of $N_1$ and $N_2$ affect the value of $S_{\text{Difference}}$?
5. Can you apply this method to a study with three or more groups?

Tip: Always double-check your calculations for rounding errors, especially when handling square roots and decimal values.