To calculate the margin of error (ME) for the 95% confidence interval, follow these steps:

1. Calculate the Differences

Compute the differences between the paired data points for Population 1 and Population 2:

$d_i = X_{1i} - X_{2i}$

The differences are: $d = [30 - 44, 15 - 28, 28 - 36, 44 - 35, 26 - 18, 47 - 35, 32 - 22]$

$d = [-14, -13, -8, 9, 8, 12, 10]$

2. Compute the Mean ($\bar{d}$) and Standard Deviation ($s_d$) of the Differences

• Mean ($\bar{d}$):

$\bar{d} = \frac{\sum d_i}{n}$

$\bar{d} = \frac{-14 + (-13) + (-8) + 9 + 8 + 12 + 10}{7} = \frac{4}{7} \approx 0.571429$

• Standard Deviation ($s_d$):

$s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}}$

1. Differences from mean: $(-14 - 0.571429)^2, (-13 - 0.571429)^2, ..., (10 - 0.571429)^2$

   Squared differences: $[211.5918, 181.7347, 73.4694, 71.0204, 55.1837, 130.6122, 88.1633]$

2. Sum of squared differences: $\sum = 811.7755$

3. Compute $s_d$: $s_d = \sqrt{\frac{811.7755}{7-1}} = \sqrt{\frac{811.7755}{6}} = \sqrt{135.2959} \approx 11.6308$

3. Determine the Critical $t$-Value

For a 95% confidence level and $n-1 = 7-1 = 6$ degrees of freedom, the $t$-value is approximately $t_{0.025, 6} = 2.447$.

4. Calculate the Margin of Error (ME)

The formula for the margin of error is:

$ME = t \cdot \frac{s_d}{\sqrt{n}}$

Substitute the values: $ME = 2.447 \cdot \frac{11.6308}{\sqrt{7}}$

1. Compute $\sqrt{7}$: $\sqrt{7} \approx 2.6458$

2. Compute $ME$: $ME = 2.447 \cdot \frac{11.6308}{2.6458} = 2.447 \cdot 4.3920 \approx 10.7526$

Final Answer:

The margin of error is approximately: $\boxed{10.752646}$

Let me know if you want the remaining steps explained or have additional questions!

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Here are 5 related questions:

1. How do you interpret a confidence interval in this context?
2. How does the assumption of normality affect the calculations?
3. What happens to the margin of error if the confidence level is increased?
4. How do you calculate the $t$-value for a different sample size?
5. Why do we use the standard error in the margin of error formula?

Tip: Always ensure the assumptions of the test (e.g., normality and dependency of samples) are valid before interpreting the results.