A box contains 250 tickets. Two people want to estimate the...

A box contains 250 tickets. Two people want to estimate the percentage of l's in the box. They agree to use the percentage of $1^{\prime}$ 's in 100 draws made at random from the box. Person A wants to draw with replacement; person B wants to draw without replacement. Which procedure gives a more accurate estimate? Or does it make any difference?

A box contains 250 tickets. Two people want to estimate the percentage of
l's in the box. They agree to use the percentage of $1^{\prime}$ 's in 100 draws made at
random from the box. Person A wants to draw with replacement; person B
wants to draw without replacement. Which procedure gives a more accurate
estimate? Or does it make any difference?

Key Concepts

Variance of an Estimator (and its Implications for Accuracy)

The accuracy of an estimate is influenced by its variance, which measures the spread or dispersion of the sampling distribution. Lower variance generally indicates a more precise estimate. When comparing different sampling methods, understanding how each method affects the variance of the sample proportion is crucial.

Sampling Methods

Sampling methods determine the process by which members are selected from a population to form a sample. This concept covers both sampling with replacement, where each selection is independent and the same unit may be chosen more than once, and sampling without replacement, where each unit is removed from the population once selected, affecting the probabilities of subsequent draws.

Independence of Observations

In sampling with replacement, each draw is independent because the composition of the population remains unchanged throughout the process. In contrast, sampling without replacement creates dependence among observations since each drawn unit alters the remaining population, which can affect the variability of the estimate.

Finite Population Correction Factor

When sampling without replacement from a finite population, the finite population correction (FPC) factor is applied to account for the reduced variability that comes from not replacing sampled units. This correction decreases the variance of the estimator when the sample size is a non-negligible fraction of the entire population, thereby potentially increasing accuracy compared to sampling with replacement.

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