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When statisticians replaced the unknown population standard deviation σ with the sample standard deviation s while working with small samples, why did simply reusing the normal-distribution method fail to produce accurate confidence intervals?
ASmall samples always have a population standard deviation of zero
BThe actual distribution of the t-score depends on the sample size
CThe sample mean cannot be computed for small samples
DConfidence intervals are undefined whenever n is below 100
Answer & Solution
Correct answer: B. The actual distribution of the t-score depends on the sample size
1. Simply substituting s for the unknown σ worked well for large samples but broke down for small ones.
2. The researcher who studied this found that the resulting distribution's shape actually depends on the sample size.
3. That discovery led to the Student's t-distribution, a family of distributions with one member for each sample size.
4. Options A, C, and D each describe conditions that are simply false about small-sample statistics.
_Source: OpenStax Introductory Statistics (CC BY 4.0), Ch 8 "Confidence Intervals", section 8.2 A Single Population Mean using the Student t Distribution_
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