Question

A sample dataset of 567 has a mean of 12.3 with a standard deviation of 0.7. What is the 95 percent confidence interval? Round the answer to the nearest hundredth. Use the table to find the critical value.

Answer 1

To calculate the 95% confidence interval for the sample mean, you'll use the following formula:

\[
\text{Confidence Interval} = \text{mean} \pm (z^* \times \frac{\sigma}{\sqrt{n}})
\]

Where:
- \(\text{mean} = 12.3\)
- \(\sigma = 0.7\) (standard deviation)
- \(n = 567\) (sample size)
- \(z^*\) is the critical value for a 95% confidence interval.

1. **Find the critical value**: For a 95% confidence interval, the critical value \(z^*\) is approximately 1.96 (this value can be found in a z-table).

2. **Calculate the standard error (SE)**:
\[
\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{0.7}{\sqrt{567}} \approx \frac{0.7}{23.8118} \approx 0.0294
\]

3. **Calculate the margin of error (ME)**:
\[
\text{ME} = z^* \times \text{SE} = 1.96 \times 0.0294 \approx 0.0576
\]

4. **Calculate the confidence interval**:
\[
\text{Lower Limit} = \text{mean} - \text{ME} = 12.3 - 0.0576 \approx 12.2424
\]
\[
\text{Upper Limit} = \text{mean} + \text{ME} = 12.3 + 0.0576 \approx 12.3576
\]

5. **Round to the nearest hundredth**:
- Lower Limit: \(12.24\)
- Upper Limit: \(12.36\)

Thus, the 95% confidence interval is approximately \((12.24, 12.36)\).