To calculate the 95 percent confidence interval for the mean, you can use the following formula:
\[ \text{Confidence Interval} = \text{Mean} \pm (z \times \frac{\sigma}{\sqrt{n}}) \]
Where:
- \(\text{Mean} = 12.3\)
- \(z\) is the critical value for a 95% confidence level, which is \(1.96\)
- \(\sigma = 0.7\) (standard deviation)
- \(n = 567\) (sample size)
First, calculate the standard error (SE):
\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.7}{\sqrt{567}} \]
Calculating \(\sqrt{567}\):
\[ \sqrt{567} \approx 23.811 \]
Now substitute this back into the formula for standard error:
\[ SE \approx \frac{0.7}{23.811} \approx 0.0294 \]
Now, calculate the margin of error (ME):
\[ ME = z \times SE = 1.96 \times 0.0294 \approx 0.0576 \]
Finally, calculate the confidence interval:
\[ \text{Lower Limit} = \text{Mean} - ME = 12.3 - 0.0576 \approx 12.2424 \]
\[ \text{Upper Limit} = \text{Mean} + ME = 12.3 + 0.0576 \approx 12.3576 \]
Rounding to the nearest hundredth, the confidence interval is:
\[ \text{Confidence Interval} \approx (12.24, 12.36) \]
So, the final 95 percent confidence interval is (12.24, 12.36).