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In a study of acupuncture, 15 subjects give a sample mean sensory rate of x-bar = 8.2267 and sample standard deviation s = 1.6722. Why must the confidence interval here use the Student's t-distribution rather than the normal distribution?
AThe sample size of 15 exceeds thirty already
BThe confidence level requested exceeds 99 percent
CThe population standard deviation is unknown, only s is given
DThe sensory rates are not numerical values at all
Answer & Solution
Correct answer: C. The population standard deviation is unknown, only s is given
1. Only a sample standard deviation, s = 1.6722, is given here, not a known population σ.
2. Whenever s must estimate an unknown σ, the correct method uses the Student's t-distribution.
3. With n = 15, the degrees of freedom are df = 15 - 1 = 14.
4. Option A is false, since 15 is smaller than 30, and that threshold would not matter under modern practice anyway.
_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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