91Ó°ÊÓ

Sampling is a fundamental concept in statistics. It involves selecting a subset, or sample, from a larger group, known as the population. Here, the population consists of the 10,000 signatures, while we take a sample of 2,000 signatures for further analysis.

This method is vital for gaining information when dealing with large datasets. Sampling allows us to make inferences about the entire population without examining every single item. This saves both time and resources.

There are several methods of sampling, such as:

• Random Sampling: Every member of the population has an equal chance of being selected.
• Systematic Sampling: Members are selected at regular intervals.
• Stratified Sampling: The population is divided into different subgroups, and samples are taken from each.

In the exercise, the sample of 2000 signatures involves random selection, which helps in maintaining the randomness required for statistical analysis.

Probability is the study of uncertainty and chance, which is essential in determining the likelihood of various outcomes. In our context, we are interested in the probability of selecting an invalid signature from the sample.

The probability is expressed as a number between 0 and 1, where 0 implies an impossible event and 1 indicates certainty. For this exercise:

• The probability of success (invalid signature) is 20% or 0.2.
• The probability of failure (valid signature) is 80% or 0.8.

Using these probabilities, we can predict the number of invalid signatures in the sample of 2,000.

Understanding probability is crucial for determining the expected outcomes and making decisions based on statistical data. It enables us to evaluate the feasibility of certain hypotheses, like the assumption of a binomial distribution in this case.

The Finite Population Correction (FPC) is a useful adjustment made when sampling from a finite population without replacement. It helps in ensuring more accurate results when the sample size is not negligible compared to the population size.

In our example with the 10,000 signatures, we sample 2,000. While this number might not be very small, it is still only a fifth of the total, which keeps the sampling influence relatively minor. However, technically, such sampling without replacement makes the trials less than perfectly independent.

Normally, to account for this lack of independence, we apply the FPC, which slightly adjusts our calculations compared to cases involving infinite populations, or where the sample size is very small. The correction factor is given by:\[ \text{FPC} = \sqrt{\frac{N-n}{N-1}} \] where \( N \) is the population size, and \( n \) is the sample size.

Applying FPC allows the approximation of the distribution as binomial, as it ensures that small deviations due to sampling without replacement do not significantly affect the analysis.