In the previous exercise, suppose that it is decided that a...

In the previous exercise, suppose that it is decided that a sample of 100 voters is too small to provide a sufficiently reliable estimate of the population proportion. It is required instead that the probability that the sample proportion differs from the population proportion (whatever its value) by more than 0.03 should not exceed 0.05 . How large a sample is needed to guarantee that this requirement is met?

Key Concepts

Normal Approximation to the Binomial Distribution

This concept involves using the normal distribution as an approximation of the binomial distribution when the sample size is large enough. It is fundamental in simplifying the calculations for confidence intervals and probability assessments regarding the sample proportion, particularly when the exact binomial calculations become cumbersome.

Worst-case Variability

In proportion estimation, the variability is maximized when the true proportion is 0.5, because the product p(1-p) reaches its maximum value of 0.25 at that point. Using this worst-case scenario ensures that computed sample sizes are conservative, guaranteeing the desired level of precision regardless of the actual proportion.

Margin of Error

The margin of error represents the maximum allowed difference between the sample estimate and the true population parameter at a given confidence level. It is a critical measure in determining the reliability of an estimate, as a smaller margin of error demands a larger sample size to ensure the estimates remain within that bound.

Sample Size Determination

This concept involves computing the number of observations needed to achieve a specified level of precision for an estimate, given a desired margin of error and level of confidence. It typically employs parameters from the normal distribution, such as the z-score, in combination with an estimate of variability to solve for the minimum sample size required.

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