Confidence Interval for Mean Using Z-Test

 

Confidence Interval for Mean

When the standard deviation of the population is known, the confidence interval for the population mean can be calculated using the Z-test. The Z-test assumes that the population follows a normal distribution or that the sample size is large enough to apply the Central Limit Theorem. Here's how you can calculate a confidence interval for the population mean with a known standard deviation:

1. Determine the confidence level: Specify the desired level of confidence for the interval. Common choices are 90%, 95%, or 99%.
2. Identify the critical value: Determine the critical value associated with the chosen confidence level. This value corresponds to the z-score from the standard normal distribution. For example, for a 95% confidence level, the critical value is approximately 1.96.
3. Gather sample data: Take a random sample from the population of interest. While the sample size doesn't affect the calculation of the confidence interval in this case, having a larger sample size generally provides more reliable results.
4. Calculate the standard error: The standard error (SE) represents the standard deviation of the sample mean and is calculated by dividing the population standard deviation (σ) by the square root of the sample size (n). The formula is SE = σ / sqrt(n).
5. Calculate the margin of error: The margin of error is the product of the critical value and the standard error. It represents the maximum distance from the sample mean that the true population mean is likely to fall within. The formula is Margin of Error = Critical Value * Standard Error.
6. Compute the confidence interval: The confidence interval is calculated by subtracting and adding the margin of error to the sample mean. The formula is Confidence Interval = Sample Mean ± Margin of Error.

Here's an example to illustrate the process:

Suppose you want to estimate the average weight of a certain population of adults. You know from previous research that the population standard deviation is 10 kg. You take a random sample of 50 individuals and find that the sample mean weight is 70 kg.

1. Choose the confidence level: Let's select a 95% confidence level.
2. Determine the critical value: For a 95% confidence level, the critical value is 1.96 (approximate value).
3. Calculate the standard error: SE = 10 / sqrt(50) ≈ 1.41 kg.
4. Calculate the margin of error: Margin of Error = 1.96 * 1.41 ≈ 2.77 kg.
5. Compute the confidence interval: Confidence Interval = 70 ± 2.77, which results in an interval of (67.23, 72.77).

Thus, based on this sample, we can say with 95% confidence that the average weight of the population lies between 67.23 kg and 72.77 kg.

It's important to note that this calculation assumes that the population standard deviation provided is accurate and that the sample is representative of the population. Additionally, if the population standard deviation is not known and needs to be estimated from the sample, a t-test, and the t-distribution are used instead of the Z-test and the standard normal distribution.