In order to compare the means of two​ populations, independent random samples of 410 observations are selected from each​ population, with the results found in the table to the right. Complete parts a through e below.
Sample 1
Sample 2
x overbar 1equals5 comma 314
x overbar 2equals5 comma 273
s 1equals148
s 2equals197
Question content area bottom
Part 1
a. Use a​ 95% confidence interval to estimate the difference between the population means left parenthesis mu 1 minus mu 2 right parenthesis. Interpret the confidence interval.
The confidence interval is

### 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.

In order to compare the means of two populations, independent random samples of 410 observations are selected from each population, with the results found in the table to the right. Complete parts a through e below. Sample 1: x̄₁ = 5,314, s₁ = 148. Sample 2: x̄₂ = 5,273, s₂ = 197. a. Use a 95% confidence interval to estimate the difference between the population means (μ₁ - μ₂). Interpret the confidence interval.

Learn how to calculate a 95% confidence interval for the difference between population means using independent samples. This example involves comparing two sample means (x̄₁ = 5,314 and x̄₂ = 5,273) with standard deviations (s₁ = 148 and s₂ = 197) and sample sizes of 410. The solution walks through each step, including finding the standard error and margin of error.

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