The formula for the interval estimate of the population mean depends on whether the population standard deviation is known or unknown.
1. When the Population Standard Deviation (σ) is Known:
The formula for the confidence interval estimate of the population mean (μ) is:
*x̄ ± z(σ/√n)**
Where:
- x̄ = Sample mean
- z = z-score corresponding to the desired confidence level (e.g., for a 95% confidence interval, z = 1.96)
- σ = Population standard deviation
- n = Sample size
Explanation:
This formula calculates a range around the sample mean (x̄) within which we are confident the true population mean (μ) lies. The z-score determines the width of the interval based on the desired level of confidence. A larger sample size (n) reduces the width of the interval, providing a more precise estimate.
Example:
Suppose a sample of 50 items has a mean of 100. The population standard deviation is known to be 15. We want to calculate a 95% confidence interval.
- x̄ = 100
- z = 1.96 (for 95% confidence)
- σ = 15
- n = 50
Confidence Interval = 100 ± 1.96 * (15/√50) ≈ 100 ± 4.16
So, the 95% confidence interval is approximately (95.84, 104.16).
2. When the Population Standard Deviation (σ) is Unknown:
When the population standard deviation is unknown, we estimate it using the sample standard deviation (s). In this case, we use the t-distribution instead of the z-distribution. The formula becomes:
*x̄ ± t(s/√n)**
Where:
- x̄ = Sample mean
- t = t-score corresponding to the desired confidence level and degrees of freedom (df = n-1)
- s = Sample standard deviation
- n = Sample size
Explanation:
The t-distribution accounts for the added uncertainty of estimating the population standard deviation. The t-score depends on the degrees of freedom (n-1), which reflects the sample size. As the sample size increases, the t-distribution approaches the z-distribution.
Example:
Suppose a sample of 25 items has a mean of 75 and a sample standard deviation of 10. We want to calculate a 99% confidence interval.
- x̄ = 75
- t = 2.797 (for 99% confidence and df = 24, from a t-table or calculator)
- s = 10
- n = 25
Confidence Interval = 75 ± 2.797 * (10/√25) ≈ 75 ± 5.594
So, the 99% confidence interval is approximately (69.406, 80.594).
In summary, the appropriate formula depends on whether the population standard deviation is known or if you must use the sample standard deviation as an estimate.
