50 Σ 8 N 400 Calculate A 99 Confidence Interval

This calculator helps you determine a 99% confidence interval for a population with a known standard deviation of 50, a sample size of 8, and a population size of 400. The confidence interval provides a range of values that is likely to contain the true population mean with 99% confidence.

How to Calculate a 99% Confidence Interval

To calculate a 99% confidence interval for a population mean when the population standard deviation is known, follow these steps:

Identify the sample mean (x̄), sample size (n), population standard deviation (σ), and population size (N).
Determine the critical value (z*) from the standard normal distribution table for a 99% confidence level.
Calculate the standard error (SE) using the formula: SE = σ / √n.
Compute the margin of error (ME) using the formula: ME = z* × SE.
Determine the confidence interval using the formula: (x̄ - ME, x̄ + ME).

Use the calculator on the right to perform these calculations quickly and accurately.

The Formula

The formula for calculating a confidence interval when the population standard deviation is known is:

Confidence Interval = (x̄ - z* × (σ / √n), x̄ + z* × (σ / √n))

Where:

x̄ = sample mean
z* = critical value from the standard normal distribution
σ = population standard deviation
n = sample size

The critical value (z*) for a 99% confidence level is approximately 2.576.

Worked Example

Let's calculate a 99% confidence interval for a population with σ = 50, n = 8, and N = 400, given a sample mean (x̄) of 100.

Identify the values: x̄ = 100, σ = 50, n = 8, z* = 2.576.
Calculate the standard error: SE = 50 / √8 ≈ 17.678.
Compute the margin of error: ME = 2.576 × 17.678 ≈ 45.6.
Determine the confidence interval: (100 - 45.6, 100 + 45.6) = (54.4, 145.6).

The 99% confidence interval for the population mean is approximately 54.4 to 145.6.

Interpreting the Result

The confidence interval provides a range of values that is likely to contain the true population mean with 99% confidence. In the example above, we can be 99% confident that the true population mean falls between 54.4 and 145.6.

If the confidence interval is wide, it indicates that the sample size is small or the population standard deviation is large, making it harder to estimate the population mean precisely. If the confidence interval is narrow, it suggests that the sample size is large or the population standard deviation is small, allowing for a more precise estimate of the population mean.

FAQ

What is a 99% confidence interval?: A 99% confidence interval is a range of values that is likely to contain the true population mean with 99% confidence. It is calculated using the sample mean, sample size, population standard deviation, and a critical value from the standard normal distribution.
When should I use a 99% confidence interval?: Use a 99% confidence interval when you need a high level of confidence in your estimate of the population mean. This is appropriate in situations where the consequences of being wrong are severe, such as in medical research or quality control.
How does sample size affect the confidence interval?: The sample size affects the width of the confidence interval. A larger sample size results in a narrower confidence interval, providing a more precise estimate of the population mean. A smaller sample size results in a wider confidence interval, indicating more uncertainty in the estimate.
What is the difference between a confidence interval and a margin of error?: The confidence interval is a range of values that is likely to contain the true population mean with a certain level of confidence. The margin of error is half the width of the confidence interval and represents the maximum expected difference between the sample estimate and the true population parameter.
Can I calculate a confidence interval without knowing the population standard deviation?: No, calculating a confidence interval for the population mean requires knowledge of the population standard deviation. If the population standard deviation is unknown, you can use the sample standard deviation to estimate it.