To calculate a 99% confidence interval, you'll need the following information:
- The sample mean (\( \bar{x} \))
- The sample standard deviation (\( s \))
- The sample size (\( n \))
- The critical value for a 99% confidence level (usually from the z-table or t-table depending on whether the population standard deviation is known and the sample size).
The formula for the confidence interval is:
\[ \text{CI} = \bar{x} \pm (z^* \times \frac{s}{\sqrt{n}}) \]
or
\[ \text{CI} = \bar{x} \pm (t^* \times \frac{s}{\sqrt{n}}) \]
Depending on the context.
Steps to calculate the confidence interval:
- Determine the sample mean (\( \bar{x} \)) and sample standard deviation (\( s \)).
- Find the critical value (\( z^* \) or \( t^* \)):
- For a 99% confidence level, the critical z-value is typically approximately 2.576 (for z-distribution).
- If using t-distribution, find the appropriate critical value from the t-table based on degrees of freedom (\( n - 1 \)).
- Calculate the standard error: \[ SE = \frac{s}{\sqrt{n}} \]
- Calculate the margin of error: \[ \text{Margin of Error} = z^* \times SE \quad \text{or} \quad t^* \times SE \]
- Determine lower and upper bounds: \[ \text{Lower Bound} = \bar{x} - \text{Margin of Error} \] \[ \text{Upper Bound} = \bar{x} + \text{Margin of Error} \]
Example Calculation:
Given:
- Sample Mean (\( \bar{x} \)): 50
- Sample Standard Deviation (\( s \)): 10
- Sample Size (\( n \)): 30
-
Calculate the standard error: \[ SE = \frac{10}{\sqrt{30}} \approx 1.8257 \]
-
Critical value for 99% confidence level (normal assumption):
- \( z^* \approx 2.576 \)
-
Calculate the Margin of Error: \[ \text{Margin of Error} = 2.576 \times 1.8257 \approx 4.699 \]
-
Calculate the confidence interval:
- Lower Bound: \[ \text{Lower Bound} = 50 - 4.699 \approx 45.301 \]
- Upper Bound: \[ \text{Upper Bound} = 50 + 4.699 \approx 54.699 \]
Final Result (Rounded to nearest thousandths):
- Lower bound of the interval: 45.301
- Upper bound of the interval: 54.699
If you provide specific values for your sample mean, standard deviation, and sample size, I can refine this calculation for you.