For independent random samples from the same population, holding variability, design and confidence level fixed, how does increasing sample size generally affect precision of the sample mean?
Correct Answer
C) Larger samples generally reduce standard error
Holding population variability, sampling design and confidence level fixed, a larger independent random sample generally reduces the standard error of its mean. This precision improvement does not remove selection bias or make incorrect models reliable. A universal minimum of 10,000 is not implied, and a median alone does not establish sampling precision.
Why This Is the Correct Answer
Option C is correct because the standard error of the sample mean is the standard deviation divided by the square root of the sample size. As the sample grows, that denominator grows, so the standard error generally shrinks and the estimate becomes more precise. This improvement does not remove selection bias or fix a flawed model.
Why the Other Options Are Wrong
Option A: Sample size has no bearing on standard error
Option A, saying sample size has no bearing on standard error, contradicts the formula: the standard error is the standard deviation divided by the square root of the sample size.
Option B: Smaller samples provide more precise estimates
Option B, saying smaller samples provide more precise estimates, reverses the relationship. Smaller samples generally produce a larger standard error and less precision.
Option D: No mean can be estimated from fewer than 10,000 observations
Option D, requiring at least 10,000 observations to estimate any mean, states a minimum that does not exist. A mean can be estimated from any sample, with precision depending on size and variability.
More data, less wobble
A bigger random sample makes the sample mean wobble less from sample to sample.
How to use: Use this to remember the direction of the sample size and standard error relationship.
Exam Tip
Precision and bias are different ideas; a larger sample improves precision but does not cure bias.
Common Mistakes to Avoid
- -Believing smaller samples are more precise
- -Thinking a large sample removes selection bias
- -Assuming there is a universal minimum sample size
Concept Deep Dive
Analysis
This question tests the relationship between sample size and precision. The standard error of the mean equals the standard deviation divided by the square root of the sample size. Holding other factors fixed, a larger sample therefore generally gives a smaller standard error.
Background Knowledge
Random sampling error affects statistical precision even when all data are accurate. The standard error quantifies that precision, and it declines as sample size increases, though bias from non-random selection is a separate problem.
Real-World Application
An appraiser estimating a typical rent in a market compares results from 8 leases and 40 leases drawn randomly from the same apartment segment. The larger set yields a tighter estimate of the mean rent, although the appraiser still checks that the leases are representative.
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An appraiser arranges twenty sale prices in ascending order: $290,000, $310,000, $330,000, $350,000, $370,000, $390,000, $410,000, $430,000, $450,000, $470,000, $490,000, $510,000, $530,000, $550,000, $570,000, $590,000, $610,000, $630,000, $650,000, $670,000. What is the 10th percentile by the nearest-rank method, which takes the lowest price at or below which at least 10% of the sales fall?
Next Question
The assessment-to-sale ratios are 0.8, 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15, 1.2. Calculate COD as [average absolute deviation from the median ÷ median]×100. For even n, average the two middle ratios. Report the COD as a percentage to two decimals.