If you increase the sample size from 25 to 100 while keeping the population standard deviation constant, what happens to the standard deviation of the sampling distribution of the mean?

Question

BlackTom AI · Accepted Answer

The standard error (SE) of the sample mean is σ/√n, so when n changes the SE changes with the square root of n.

1) Relevant concepts:
- Population standard deviation: σ (held constant here).
- Standard deviation of the sampling distribution of the mean (standard error): SE = σ/√n.
- Larger n → smaller SE → greater precision (narrower sampling distribution).

2) Step-by-step reasoning and 3) calculations:
- For the original sample size n1 = 25: SE1 = σ / √25 = σ / 5.
- For the larger sample size n2 = 100: SE2 = σ / √100 = σ / 10.
- Compute the factor of change: SE2 / SE1 = (σ/10) / (σ/5) = (1/10) / (1/5) = (1/10) × (5/1) = 1/2.
Thus the standard deviation of the sampling distribution is reduced to one-half of its original value.

4) Why this answer is correct:
Because SE depends on the square root of n, quadrupling the sample size (from 25 to 100) multiplies √n by 2, so SE is divided by 2. A smaller SE means the sampling distribution of the mean is narrower, giving greater precision in estimating the population mean.

5) Why the other options are incorrect:
- b (It doubles): wrong sign — SE decreases when n increases.
- c (decreases to one-fourth): confuses variance with standard deviation. The sampling variance (σ^2/n) would become one-fourth, but the standard deviation (SE) becomes one-half.
- d (increases by factor of 4): opposite of reality.
- e (decreases by half, making the distribution wider): internal contradiction — decreasing the SE makes the distribution narrower, not wider.

6) Conclusion:
The standard deviation of the sampling distribution decreases by half, increasing precision; this confirms the correct choice is option a.

If you increase the sample size from 25 to 100 while keeping the population standard deviation constant, what happens to the standard deviation of the sampling distribution of the mean?单项选择题

类似问题

What is a sampling distribution?

Which of the following statements is correct about a parameter or a statistic associated with repeated random samples of the same size from the same population?

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