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.
-
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).
-
Calculate the standard error (SE): \[ \text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{0.7}{\sqrt{567}} \approx \frac{0.7}{23.8118} \approx 0.0294 \]
-
Calculate the margin of error (ME): \[ \text{ME} = z^* \times \text{SE} = 1.96 \times 0.0294 \approx 0.0576 \]
-
Calculate the confidence interval: \[ \text{Lower Limit} = \text{mean} - \text{ME} = 12.3 - 0.0576 \approx 12.2424 \] \[ \text{Upper Limit} = \text{mean} + \text{ME} = 12.3 + 0.0576 \approx 12.3576 \]
-
Round to the nearest hundredth:
- Lower Limit: \(12.24\)
- Upper Limit: \(12.36\)
Thus, the 95% confidence interval is approximately \((12.24, 12.36)\).