Construct a confidence interval for the mean of the paired differences for the two populations with the following information. Round the endpoints of the interval to three decimal places, if necessary.

n=31, d¯=2.538, α=0.05, sd=0.556

To construct a confidence interval for the mean of the paired differences, we can use the formula for the confidence interval of a mean with a t-distribution, since the population standard deviation is unknown and the sample size is small.

The formula for the confidence interval is:

\[
\bar{d} \pm t_{\alpha/2, \, n-1} \left( \frac{s_d}{\sqrt{n}} \right)
\]

Where:
- \(\bar{d}\) is the mean of the paired differences (2.538)
- \(s_d\) is the standard deviation of the paired differences (0.556)
- \(n\) is the sample size (31)
- \(t_{\alpha/2, \, n-1}\) is the critical value from the t-distribution for a confidence level of \(1 - \alpha\) with \(n-1\) degrees of freedom.

### Step 1: Determine the \(t\)-critical value
For a 95% confidence level (\(\alpha = 0.05\)), the \(t\)-critical value can be found using a t-distribution table or calculator with \(n-1 = 30\) degrees of freedom. The critical value for \(t_{0.025, \, 30}\) is approximately 2.042.

### Step 2: Compute the margin of error
Calculate the margin of error using:

\[
\text{Margin of Error} = t_{0.025, \, 30} \left( \frac{s_d}{\sqrt{n}} \right) = 2.042 \left( \frac{0.556}{\sqrt{31}} \right)
\]

\[
= 2.042 \left( \frac{0.556}{5.568} \right) \approx 2.042 \times 0.0998 \approx 0.204
\]

### Step 3: Construct the confidence interval
Now, calculate the confidence interval as:

\[
2.538 \pm 0.204
\]

This gives the endpoints:

- Lower endpoint: \(2.538 - 0.204 = 2.334\)
- Upper endpoint: \(2.538 + 0.204 = 2.742\)

### Final Confidence Interval
The 95% confidence interval for the mean of the paired differences is:

\[
(2.334, \, 2.742)
\]

Would you like further details on the calculations, or do you have any questions?

Here are five related questions to expand your understanding:
1. How would the confidence interval change if the sample size were larger?
2. What happens to the margin of error if the standard deviation increases?
3. How does changing the confidence level (e.g., to 90% or 99%) affect the interval?
4. Why do we use the t-distribution instead of the z-distribution here?
5. What does the confidence interval suggest about the true mean of the paired differences?

**Tip:** The width of a confidence interval depends on the sample size, variability in the data, and the confidence level chosen. Increasing the sample size reduces the interval width.

Construct a confidence interval for the mean of the paired differences for the two populations with the following information. Round the endpoints of the interval to three decimal places, if necessary.
n=31, d¯=2.538, α=0.05, sd=0.556

Learn how to construct a 95% confidence interval for the mean of paired differences with n=31, a mean of d̄=2.538, and standard deviation sd=0.556. This step-by-step solution uses the t-distribution to calculate the interval for small samples. Ideal for college-level statistics.

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