Question
Find the 99% confidence interval for your sample.
Lower bound of the interval is
Upper bound of the interval is
Please round your answers to nearest thousandths.
Lower bound of the interval is
Upper bound of the interval is
Please round your answers to nearest thousandths.
Answers
GPT-4o mini
To calculate a 99% confidence interval, you'll need the following information:
1. The sample mean (\( \bar{x} \))
2. The sample standard deviation (\( s \))
3. The sample size (\( n \))
4. 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:
1. **Determine the sample mean (\( \bar{x} \))** and sample standard deviation (\( s \)).
2. **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 (\( n - 1 \)).
3. **Calculate the standard error**:
\[
SE = \frac{s}{\sqrt{n}}
\]
4. **Calculate the margin of error**:
\[
\text{Margin of Error} = z^* \times SE \quad \text{or} \quad t^* \times SE
\]
5. **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
1. **Calculate the standard error**:
\[
SE = \frac{10}{\sqrt{30}} \approx 1.8257
\]
2. **Critical value for 99% confidence level** (normal assumption):
- \( z^* \approx 2.576 \)
3. **Calculate the Margin of Error**:
\[
\text{Margin of Error} = 2.576 \times 1.8257 \approx 4.699
\]
4. **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.
1. The sample mean (\( \bar{x} \))
2. The sample standard deviation (\( s \))
3. The sample size (\( n \))
4. 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:
1. **Determine the sample mean (\( \bar{x} \))** and sample standard deviation (\( s \)).
2. **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 (\( n - 1 \)).
3. **Calculate the standard error**:
\[
SE = \frac{s}{\sqrt{n}}
\]
4. **Calculate the margin of error**:
\[
\text{Margin of Error} = z^* \times SE \quad \text{or} \quad t^* \times SE
\]
5. **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
1. **Calculate the standard error**:
\[
SE = \frac{10}{\sqrt{30}} \approx 1.8257
\]
2. **Critical value for 99% confidence level** (normal assumption):
- \( z^* \approx 2.576 \)
3. **Calculate the Margin of Error**:
\[
\text{Margin of Error} = 2.576 \times 1.8257 \approx 4.699
\]
4. **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.