Which of the following statements about sampling error are correct?
(A) The sampling error of the mean is \(\bar{x} - \mu\), where \(\bar{x}\) is the sample mean and \(\mu\) is the population mean.
(B) A sampling error can be negative.
(C) A sample chosen by simple random sampling has no sampling error.
(D) The sampling error generally decreases as the sample size increases.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
The difference between a sample statistic and the population parameter is the sampling error. For the mean,
Sampling error \(= \bar{x} - \mu\)
A sample, even a random one, is not identical to the population, so this difference is usually not zero. It is negative whenever \(\bar{x} \lt \mu\).
So the sampling error is \(\bar{x} - \mu\), it can be positive or negative, it occurs even in random samples, and it becomes smaller as the sample grows.
Checking the statements
(A) The sampling error of the mean is \(\bar{x} - \mu\), where \(\bar{x}\) is the sample mean and \(\mu\) is the population mean. — Correct. It is the sample mean minus the population mean.
(B) A sampling error can be negative. — Correct. It is negative when the sample mean is below \(\mu\).
(C) A sample chosen by simple random sampling has no sampling error. — Incorrect. A random sample still differs from the population, so it still has sampling error.
(D) The sampling error generally decreases as the sample size increases. — Correct. A larger sample is closer to the whole population.