Which statement is most nearly correct, other things being equal?
A. Doubling the sample size will cut the standard error of the mean in half.
B. The standard error of the mean depends on the population size.
C. Quadrupling the sample size roughly halves the standard error of the mean.
D. The standard error of the mean depends on the confidence level.
The correct answer and explanation is :
The correct answer is A. Doubling the sample size will cut the standard error of the mean in half.
Explanation:
The standard error of the mean (SEM) is a measure of how much the sample mean (average) is expected to vary from the true population mean. It is calculated as:
[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
]
where:
- (\sigma) is the population standard deviation, and
- (n) is the sample size.
Now, let’s go through each statement to understand why A is the correct choice:
A. Doubling the sample size will cut the standard error of the mean in half.
- True: The formula for SEM shows that it is inversely proportional to the square root of the sample size. This means that if you increase the sample size, the standard error will decrease. Specifically, if you double the sample size ((n)), the SEM is reduced by a factor of (\sqrt{2}) (approximately 1.41), which is close to halving the standard error. Therefore, doubling the sample size reduces the standard error by about 30%, but this still makes this statement the most nearly correct.
B. The standard error of the mean depends on the population size.
- False: The SEM does not directly depend on the population size. It depends on the population standard deviation ((\sigma)) and the sample size ((n)). The SEM is primarily concerned with how accurately a sample mean estimates the population mean, which is influenced by the sample size, not the overall population size. When the population size is large relative to the sample size, this can be true in finite population correction situations, but generally, SEM does not depend on the total population size.
C. Quadrupling the sample size roughly halves the standard error of the mean.
- False: Quadrupling the sample size ((n \to 4n)) will reduce the SEM by a factor of (\sqrt{4} = 2), which halves the standard error. This is true, but it’s less precise than option A. Quadrupling the sample size leads to a larger reduction in SEM than doubling the sample size, but this isn’t as accurate for the given comparison.
D. The standard error of the mean depends on the confidence level.
- False: The SEM itself does not depend on the confidence level. However, confidence intervals do depend on the confidence level (i.e., 95% or 99%). The confidence level determines the multiplier (z-score or t-score) used when calculating a confidence interval, but this multiplier does not affect the standard error directly.
In conclusion, statement A is the most accurate, as the standard error decreases in relation to the square root of the sample size, and doubling the sample size will approximately cut the SEM in half.