A clinical psychologist claims that as social support increases this causes life satisfaction to increase.

They want to test this experimentally by having 6 students receiving additional social support in their fall semester and then assessing life satisfaction of participants ( = mu1). Then the same 6 students receive no additional social support in their spring semester and are assessed on their life satisfaction during finals ( = mu2) and they want to assess if the difference in stress with no additional finals relative to additional finals with mu1 - mu2.

 = mu1 = Additional social support and 

 = mu2 = No additional social support
 
They get the following data:

**Additional social support**

**No Additional social support**

86

70

81

47

82

83

78

69

22

53

96

79

What would be the estimated standard error of the differences (i.e., the denominator of your t-statistic)? Please compute and report to 4 decimal places.

To calculate the estimated standard error of the differences (\(SE_d\)) between the two conditions, we follow these steps:

### Step 1: Calculate the Differences (\(d\))
Since we have repeated measures, we will find the difference for each student between their scores under the two conditions: \(\text{Additional social support}\) (\(X_1\)) and \(\text{No additional social support}\) (\(X_2\)):

\[
d_i = X_{1i} - X_{2i}
\]

### Step 2: Compute the Mean of the Differences (\(\bar{d}\))
The mean difference \(\bar{d}\) is calculated as:

\[
\bar{d} = \frac{\sum d_i}{n}
\]

where \(n\) is the number of pairs (which is 6 in this case).

### Step 3: Calculate the Sum of Squared Differences (\(\sum (d_i - \bar{d})^2\))
To compute the standard error, we first need the sum of squared deviations of the differences from their mean.

### Step 4: Calculate the Estimated Standard Error of the Differences (\(SE_d\))
The formula for the estimated standard error of the differences is:

\[
SE_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n (n-1)}}
\]

### Now, let’s calculate each step:

1. Calculate the differences (\(d\)) between the two sets of scores.
2. Compute the mean of these differences.
3. Find the sum of squared deviations.
4. Calculate the standard error using the formula.

Let's compute this.The estimated standard error of the differences (\(SE_d\)) is approximately **8.9839** when rounded to four decimal places.

Would you like a detailed breakdown of these calculations or have any other questions?

Here are 5 related questions that could help deepen your understanding:
1. How do you interpret the result of a t-test using the standard error?
2. What are the assumptions of a paired t-test in this experimental design?
3. How does the standard error influence the t-statistic value?
4. Why is it important to use paired data when comparing conditions like this?
5. How would the calculation change if there were more participants?

**Tip:** The standard error measures how much the sample mean differences are expected to vary due to random sampling, which is crucial for determining if observed differences are statistically significant.

A clinical psychologist claims that as social support increases this causes life satisfaction to increase. They want to test this experimentally by having 6 students receiving additional social support in their fall semester and then assessing life satisfaction of participants (mu1). Then the same 6 students receive no additional social support in their spring semester and are assessed on their life satisfaction during finals (mu2) and they want to assess if the difference in stress with no additional finals relative to additional finals with mu1 - mu2. mu1 = Additional social support and mu2 = No additional social support. They get the following data.

Learn how to calculate the estimated standard error of differences (SE_d) for paired data in a clinical psychology experiment. This example includes data on life satisfaction with additional and no additional social support. Follow detailed steps using key statistical concepts and formulas to understand the calculations.

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