Final answer:
The 68.3% confidence interval for the mean age, when taking a sample of 400 names with a mean age of 34.8 years and a sample standard deviation of 10.8 years, ranges from 34.26 to 35.34 years.
Step-by-step explanation:
The question involves calculating the 68.3% confidence interval for the mean age of a sample taken from a population. We use the sample mean, sample standard deviation, and the size of the sample to estimate this interval. The relevant formula for a confidence interval when the population standard deviation is unknown and the sample size is large is:
CI = μ ± (z * (σ/√n))
Where μ is the sample mean, σ is the sample standard deviation, n is the sample size, and z is the z-score corresponding to the desired confidence level. For a 68.3% confidence level, the z-score is 1 because this is one standard deviation away from the mean in a normal distribution, which corresponds to approximately 68.3% of the data within one standard deviation from the mean (according to the empirical rule).
CI = 34.8 ± (1 * (10.8/√400))
CI = 34.8 ± (1 * (10.8/20))
CI = 34.8 ± (1 * 0.54)
CI = 34.8 ± 0.54
The 68.3% confidence interval for the mean age is therefore 34.8 years ± 0.54 years, which means it ranges from 34.26 to 35.34 years.