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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.

1 answer

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.