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For a Student's t-distribution with n - 1 degrees of freedom, why is the number of degrees of freedom exactly n - 1 rather than n?
AOne data value must always be discarded before analysis
BThe formula for s only uses n-1 of the observations
CDegrees of freedom always equal the confidence level
DThe last deviation is determined once the other n-1 are known
Answer & Solution
Correct answer: D. The last deviation is determined once the other n-1 are known
1. Computing the sample standard deviation s uses n deviations of the form (x - x-bar).
2. Because those deviations must sum to zero, the last one is fixed once the other n-1 are known.
3. Only n-1 of the deviations are free to vary, so df = n - 1.
4. Option B is close but misdescribes the mechanism, since all n observations are used to compute s, just with one constraint linking them.
_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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