Estimating Fraction of Population in Favor of a Proposition

Consider sampling from a finite population to estimate the fraction favoring a proposition using hypergeometric distribution. Explore the mean, variance, and computations involved in determining the estimator's properties. Gain insights into the estimation process and its nuances.

• Estimation
• Population
• Sampling
• Hypergeometric
• Computation

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Uploaded on Feb 16, 2025 | 0 Views

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Presentation Transcript

1. Probabilistic Models Example 2.34

2. Sampling from a Finite Population { The Hypergeometric } Consider a population of N individuals, some of whom are in favor of a certain proposition. In particular suppose that Np of them are in favor and N Np are opposed. p is assumed to be unknown. We are interested in estimating p. P is the fraction of the population that is in favor, by randomly choosing and then determining the positions of n members of the population.

3. Hence, if we let ? Usual estimator of P is ?=0 its mean and variance. The mean : ? ??/?. Let us now compute ? ? ?? = ? ?? = ?[1?? + 0? 1 ? ] = ?? ?=1 ?=0

4. Variance: ? ? ??? ?? = var ?? + 2 ??? ??,?? ?=1 ?=1 ?<? 2 - ? ?? 2 ??? ?? = ? ?? = ? 12 ? + 0 1 ? 1 ? + 0(1 ?)2 = ? ?2 = ?(1 ?)

5. ??? ??,?? = ? ???? ? ??? ?? = ? ??= 1,??= 1 ?2 = ? ??= 1 ??= 1 ? ??= 1 ?2 =(?? 1) ? 1 =?(?? 1) ?? ?2 ? ?2 ? 1

6. ? ??1 ? 1 ? ?? 1 ? 1 ?? = ?? 1 ? + 2? ? ?2 ??? ?=1 = ?? 1 ? 2 ?(? 1)(? 2)! 2 ?2 2! ? 2 ! ? ?? 1 ? 1 ?2 = ?? 1 ? ?(? 1) ? ?? 1 (? 1)?2 ? 1 = ?? 1 ? ?(? 1) ??2 ? ??2+?2 ? 1 =?? 1 ? ?(? 1) =?? 1 ? ? ? 1 ?(1 ?) ? 1

7. ?? ?=P ? ? ?=1 ?? ?=1 =? 1 ? ? 1 ?(1 ?) ?(? 1) ? ? ??? ?=1 ?2??? ?=1 ?? ?