A properly drawn random sample of one thousand individuals is used to estimate a population mean. The sampling error for the sample mean is roughly plus or minus what percent of the population mean (assuming a normal distribution and no prior knowledge of population standard deviation)?

• A. $20 \mathrm{percent}$
• B. $1 \mathrm{percent}$
• C. $10 \mathrm{percent}$
• D. $3 \mathrm{percent}$

Answer: (D)

Verified step by step guidance

1. Understand that the problem asks for the approximate sampling error (margin of error) as a percentage of the population mean when estimating the mean from a sample of size 1000, assuming a normal distribution and no prior knowledge of the population standard deviation.
2. Recall that when the population standard deviation is unknown and the sample size is large (n = 1000), the sampling distribution of the sample mean can be approximated by a normal distribution, and the standard error (SE) of the sample mean is estimated by \(SE = \frac{s}{\sqrt{n}}\), where \(s\) is the sample standard deviation.
3. Since the population mean and standard deviation are unknown, and no specific values are given, consider the relative standard error, which is the standard error divided by the population mean, expressed as a percentage: \(\text{Relative SE} = \frac{SE}{\mu} \times 100\%\).
4. Use the empirical rule or confidence interval concepts: for a 95% confidence interval, the margin of error is approximately \$1.96 \times SE$. This margin of error represents the sampling error, which can be expressed as a percentage of the population mean.
5. Given the large sample size (1000), the sampling error tends to be small. The problem's correct answer (3%) suggests that the margin of error is roughly 3% of the population mean, which aligns with the typical precision achievable with such a sample size under these assumptions.