igure Upper S Subscript Difference for each of the following studies.
Upper N 1
Upper S Subscript 1 Superscript 2
Upper N 2
Upper S Subscript 2 Superscript 2
​(a)
40
2
60
5
​(b)
50
1
30
3
​(c)
40
5
50
2
​(d)
20
3
60
3
​(e)
30
5
40
1
Question content area bottom
Part 1
Figure Upper S Subscript Difference for study​ (a). 
Upper S Subscript Differenceequals
  
enter your response here ​(Round to three decimal places as​ needed.

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).

---

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.

Figure Upper S Subscript Difference for each of the following studies. Upper N 1 Upper S Subscript 1 Superscript 2 Upper N 2 Upper S Subscript 2 Superscript 2 ​(a) 40 2 60 5 ​(b) 50 1 30 3 ​(c) 40 5 50 2 ​(d) 20 3 60 3 ​(e) 30 5 40 1. Question content area bottom Part 1 Figure Upper S Subscript Difference for study​ (a). Upper S Subscript Difference equals ​  enter your response here ​(Round to three decimal places as​ needed.

Learn how to calculate the standard error difference (S_Difference) for studies with two sample groups using pooled variance. In this example, we use data from five studies, with sample sizes and variances provided. The calculation is broken down step-by-step, rounding results to three decimal places. This problem is suitable for undergraduate statistics students.

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