Given Information:

• $\bar{x}_1 = 5,314$ (Sample 1 mean)
• $\bar{x}_2 = 5,273$ (Sample 2 mean)
• $s_1 = 148$ (Sample 1 standard deviation)
• $s_2 = 197$ (Sample 2 standard deviation)
• $n_1 = n_2 = 410$ (sample sizes for both populations)

We are tasked with finding a 95% confidence interval for the difference between the population means $(\mu_1 - \mu_2)$.

Step 1: Formula for Confidence Interval of Difference of Means

Since the two samples are independent, the formula for the confidence interval for the difference between two population means is given by:

$(\bar{x}_1 - \bar{x}_2) \pm Z_{\alpha/2} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Where:

• $\bar{x}_1 - \bar{x}_2$ is the difference between the sample means.
• $Z_{\alpha/2}$ is the critical value for the confidence level (for 95%, $Z_{\alpha/2} = 1.96$).
• $s_1^2 / n_1$ and $s_2^2 / n_2$ are the variances divided by the sample sizes.

Step 2: Compute the Difference in Sample Means

$\bar{x}_1 - \bar{x}_2 = 5,314 - 5,273 = 41$

Step 3: Compute the Standard Error (SE) of the Difference

$SE = \sqrt{\frac{148^2}{410} + \frac{197^2}{410}}$

Let me compute this.The standard error (SE) of the difference between the two sample means is approximately $12.17$.

Step 4: Calculate the Margin of Error

The margin of error (ME) is given by: $ME = Z_{\alpha/2} \times SE = 1.96 \times 12.17$

Let me calculate this.The margin of error (ME) is approximately $23.85$.

Step 5: Compute the Confidence Interval

Now, we can calculate the confidence interval for the difference in population means:

$(\bar{x}_1 - \bar{x}_2) \pm ME = 41 \pm 23.85$

Thus, the confidence interval is:

$(41 - 23.85, 41 + 23.85) = (17.15, 64.85)$

Interpretation:

We are 95% confident that the true difference between the population means $(\mu_1 - \mu_2)$ lies between 17.15 and 64.85.

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Would you like further details or have any questions?

Here are 5 related questions to explore this topic further:

1. How does changing the confidence level affect the confidence interval?
2. What assumptions are made when constructing a confidence interval for the difference in means?
3. How would increasing the sample sizes affect the confidence interval width?
4. What is the interpretation of a confidence interval that includes zero?
5. How do you interpret the result if one standard deviation is significantly larger than the other?

Tip: The width of the confidence interval is influenced by both the sample size and variability. Larger samples tend to result in narrower intervals.