A polling agency wants to determine the sample size required...

A polling agency wants to determine the sample size required to get a margin of error of no more than 3 percentage points $(0.03)$. Assume the pollsters are using a $95 \%$ confidence level. How large a sample should they take? See above for the

Key Concepts

Sample Size Determination

Sample size determination involves calculating the number of observations required to achieve a desired margin of error at a specified confidence level. The formula for estimating the sample size when dealing with proportions uses the critical value, the margin of error, and the variability of the proportion. By substituting these values, one can solve for the minimum necessary sample size to ensure reliable and valid statistical results.

Population Proportion Variability

The variability of a population proportion is quantified by the product p(1-p), where p is the estimated proportion. When this parameter is unknown, using p = 0.5 maximizes p(1-p) at 0.25, providing a conservative estimate for the sample size required. This approach ensures that the sample size is sufficient regardless of the actual proportion.

Margin of Error

The margin of error represents the maximum expected difference between the true population parameter and a sample estimate. It determines how precise the estimation is and is directly related to the critical value and sample variability. In the context of polling, a smaller margin of error requires a larger sample size to ensure the estimate is closer to the true value.

Confidence Level

The confidence level reflects the degree of certainty that the true population parameter lies within the confidence interval computed from the sample. A 95% confidence level means that if many samples were taken, about 95% of the computed intervals would contain the true value. This level is associated with a specific critical value from the standard normal distribution.

Critical Value (z-score)

The critical value, often denoted as z, corresponds to the desired confidence level in a normally distributed variable. For a 95% confidence level, the critical value is approximately 1.96. This value is used in the formula to calculate the margin of error and, in turn, the required sample size.

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