A study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of 34 residents and found the mean weight to be 181 pounds with a standard deviation of 36 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, using a sample mean and standard deviation, we follow similar steps as before. However, since the sample size is smaller (less than 30), it is more appropriate to use the t-distribution instead of the normal distribution. But given that the problem asks to use the normal distribution/empirical rule, we'll proceed with that approach.

### Step 1: Identify the Given Information
- **Sample mean (\(\bar{x}\))**: 181 pounds
- **Sample size (\(n\))**: 34
- **Sample standard deviation (\(s\))**: 36 pounds

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

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

Plugging in the values:

\[
\text{SEM} = \frac{36}{\sqrt{34}}
\]

Let's calculate this value:

\[
\text{SEM} \approx \frac{36}{5.8310} \approx 6.2
\]

### 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:

\[
\text{ME} = 1.96 \times 6.2 = 12.152
\]

### 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:

\[
\text{Lower limit} = 181 - 12.152 = 168.848
\]

\[
\text{Upper limit} = 181 + 12.152 = 193.152
\]

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

- **Lower limit**: 168.8
- **Upper limit**: 193.2

### Final 95% Confidence Interval
The 95% confidence interval for the mean weight of the residents is **(168.8, 193.2)** pounds.

Would you like more details or have any questions?

### Related Questions:
1. How would the confidence interval change if the sample standard deviation was smaller?
2. Why is it more accurate to use the t-distribution for small sample sizes?
3. What does a 95% confidence interval mean in practical terms?
4. How can sample size affect the width of the confidence interval?
5. If the mean weight of another sample from the same town was 185 pounds, how would the two samples compare?

### Tip:
For small sample sizes, especially less than 30, it's generally more accurate to use the t-distribution to account for the increased variability.

Learn how to estimate a 95% confidence interval for the mean weight of a town's residents based on a sample of 34 residents with a mean weight of 181 pounds and a standard deviation of 36 pounds. Includes step-by-step calculations using the normal distribution/empirical rule. Suitable for advanced high school students studying statistics.

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