Final answer:
To construct a 95% confidence interval for a population mean with a known standard deviation, use the formula (x - EBM, x + EBM), where x is the sample mean, and EBM is the error bound calculated using the Z-value for the desired confidence level. In the provided example, the 95% confidence interval for the population mean is (12.63, 18.17).
Step-by-step explanation:
Constructing a 95% Confidence Interval for the Population Mean
To construct a 95% confidence interval for the population mean μ when the population standard deviation σ is known, and we have a sample mean μ, the formula is (x − EBM, x + EBM), where x is the sample mean and EBM is the error bound for the mean. The error bound can be calculated using the standard normal distribution (Z-distribution) because the population standard deviation is known.
Let's consider the provided example:
- Sample mean (μ) = 15.4
- Population standard deviation (σ) = 9.0
- Sample size (n) = 40
- Confidence level (C) = 95%
Firstly, we need to find the Z-value that corresponds to the 95% confidence level. Since the normal distribution is symmetrical, we look for the Z-value that leaves 2.5% in the tail, which is approximately 1.96. Next, we calculate the error bound using the formula:
EBM = Z * (σ / √ n)
Substituting the values we get:
EBM = 1.96 * (9 / √ 40) = 1.96 * (9 / 6.32) ≈ 2.77
Therefore, the confidence interval is:
(15.4 - 2.77, 15.4 + 2.77) = (12.63, 18.17)
We can say with 95% confidence that the true population mean lies within the interval (12.63, 18.17).