A sample of 53 night-school students' ages is obtained in order to estimate the mean age of night-school students. = 25.7 years. The population variance is 12.
(a) Give a point estimate for mu.
(b) Find the 95% confidence interval for mu (lower and upper limit).
(c) Find the 99% confidence interval for mu (lower and upper limit).
For (b) I ended up with 22.47 and 28.93 and for (c) I ended up with 21.45 and 29.95.
To find the answers for parts (a), (b), and (c), we need to use the formulas for confidence intervals. The formulas for confidence intervals depend on the sample mean, sample size, population standard deviation, and the desired level of confidence.
(a) Point Estimate for μ:
The point estimate for the population mean μ is the sample mean, which is denoted by x̄. In this case, the sample mean is given as x̄ = 25.7 years.
(b) 95% Confidence Interval for μ:
To find the 95% confidence interval for μ, we need to use the formula:
Lower Limit = x̄ - Z * (σ / √n)
Upper Limit = x̄ + Z * (σ / √n)
Where:
x̄ is the sample mean
Z is the Z-score associated with a 95% confidence level (Z = 1.96)
σ is the population standard deviation (given as √12 = 3.46)
n is the sample size (given as 53)
Plugging in the values into the formula:
Lower Limit = 25.7 - 1.96 * (3.46 / √53) ≈ 22.47
Upper Limit = 25.7 + 1.96 * (3.46 / √53) ≈ 28.93
Therefore, the 95% confidence interval for μ is approximately (22.47, 28.93).
(c) 99% Confidence Interval for μ:
To find the 99% confidence interval for μ, we need to use the same formula with a different Z-score.
Lower Limit = x̄ - Z * (σ / √n)
Upper Limit = x̄ + Z * (σ / √n)
Where:
x̄ is the sample mean
Z is the Z-score associated with a 99% confidence level (Z = 2.57)
σ is the population standard deviation (given as √12 = 3.46)
n is the sample size (given as 53)
Plugging in the values into the formula:
Lower Limit = 25.7 - 2.57 * (3.46 / √53) ≈ 21.45
Upper Limit = 25.7 + 2.57 * (3.46 / √53) ≈ 29.95
Therefore, the 99% confidence interval for μ is approximately (21.45, 29.95).
Please note that rounding errors may occur, so the values provided may slightly differ from your calculations.