A study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of 86 residents and found the mean weight to be 182 pounds with a standard deviation of 28 pounds. Use the normal distribution/empirical rule to estimate a 95% confidence interval for the mean, rounding all values to the nearest tenth.

To estimate a 95% confidence interval for the mean weight of the residents in the town, we can use the sample mean and the standard deviation provided along with the normal distribution. Here’s how we calculate it step-by-step:

### Step 1: Identify the Given Information
- **Sample mean (\(\bar{x}\))**: 182 pounds
- **Sample size (\(n\))**: 86
- **Sample standard deviation (\(s\))**: 28 pounds

### Step 2: Calculate the Standard Error of the Mean (SEM)
The standard error of the mean (SEM) is given by the formula:

\[
\text{SEM} = \frac{s}{\sqrt{n}}
\]

Plugging in the values:

\[
\text{SEM} = \frac{28}{\sqrt{86}}
\]

Let's calculate this value.

\[
\text{SEM} \approx \frac{28}{9.2736} \approx 3.0
\]

### Step 3: Determine the Z-Score for a 95% Confidence Level
For a 95% confidence level, the Z-score corresponding to the normal distribution is approximately **1.96**.

### Step 4: Calculate the Margin of Error (ME)
The margin of error is calculated using the formula:

\[
\text{ME} = Z \times \text{SEM}
\]

Substituting the values:

\[
\text{ME} = 1.96 \times 3.0 = 5.88
\]

### Step 5: Construct the Confidence Interval
The 95% confidence interval is calculated using the formula:

\[
\text{Confidence Interval} = \bar{x} \pm \text{ME}
\]

Applying the values:

\[
\text{Lower limit} = 182 - 5.88 = 176.12
\]

\[
\text{Upper limit} = 182 + 5.88 = 187.88
\]

### Step 6: Round to the Nearest Tenth
Rounding the confidence interval limits to the nearest tenth:

- **Lower limit**: 176.1
- **Upper limit**: 187.9

### Final 95% Confidence Interval
The 95% confidence interval for the mean weight of the residents is **(176.1, 187.9)** pounds.

Would you like more details or have any questions?

### Related Questions:
1. How would the confidence interval change if the sample size was smaller?
2. What is the interpretation of the 95% confidence interval in this context?
3. How does changing the confidence level to 99% affect the interval?
4. What assumptions must be met for this confidence interval to be valid?
5. How can we compute a confidence interval if the data is not normally distributed?

### Tip:
Always check if the conditions for using a confidence interval (such as the sample size and normality of the data) are satisfied before applying the formula!

Learn how to estimate a 95% confidence interval for the mean weight of a town's residents based on a sample mean, sample size, and standard deviation. Explore step-by-step calculations using normal distribution principles. Suitable for advanced high school students.

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