Problem 34 Use the given data to find the m... [FREE SOLUTION] (2024)

Short Answer

population proportion

Population proportion refers to the fraction of a population that has a particular attribute or characteristic. For example, if we want to find out how many adults believe in astrology, the population proportion will determine what percentage of the entire population believes in it. Knowing this proportion helps in making predictions and creating statistical estimates about the whole population based on a sample. When the proportion is unknown, a common practice is to assume it to be 0.5 (or 50%) for the worst-case scenario, which maximizes the required sample size to ensure a conservative estimate.

confidence level

The confidence level is a measure of how certain we are about our estimate. It is expressed as a percentage and typically chosen between 90%, 95%, and 99%. The higher the confidence level, the more certain we are that our estimate includes the true population parameter. For instance, with a 99% confidence level, we are 99% confident that the population proportion falls within the calculated margin of error. This high level of confidence translates into a higher Z-value, which impacts the sample size required.

margin of error

The margin of error represents the range within which we expect the true population parameter to lie. It is usually expressed as a percentage and corresponds to the maximum amount by which we believe our sample estimate may differ from the true population proportion. For instance, if we set our margin of error at 4 percentage points, it means our estimate could be plus or minus 4% off from the actual population proportion. A smaller margin of error indicates a more precise estimate and typically requires a larger sample size.

Z-value

The Z-value is a critical component in estimating sample size, derived from the standard normal distribution corresponding to the selected confidence level. For a confidence level of 99%, the Z-value is approximately 2.576. This value indicates how many standard deviations away from the mean we should be to achieve our desired confidence level. The Z-value is squared and multiplied by the estimated proportion and its complement in the sample size formula. Higher Z-values increase the sample size to ensure greater confidence in the estimate. The formula for sample size is: \[ n = \frac{Z^2 \times p \times (1 - p)}{E^2} \]Utilizing the correct Z-value ensures that your sample size aligns accurately with your confidence level requirements.