Question

What sample size is needed to give the desired margin of error in estimating a population proportion with the indicated level of confidence? A margin of error within \(\pm 3 \%\) with \(90 \%\) confidence. We estimate that the population proportion is about \(0.3 .\)

Short answer

The sample size needed to give the desired margin of error within \(\pm 3 \%\) with \(90 \%\) confidence is \(845\).

Step-by-step solution

Identifying key values

The margin of error (E) is \(\pm 3 \%\), which translates to \(\pm 0.03\) in decimal form. The level of confidence is \(90 \%\), implying a Z-score (Z) of \(1.645\), which aligns with standard Z-scores for given confidence levels. We also know that the estimated population proportion (p) is \(0.3\).

Applying the formula to find the sample size

Inserting the values E = 0.03, Z = 1.645 and p = 0.3 into the formula \(n = \frac{Z^2 * p * (1 - p)}{E^2}\), the required solution becomes \(n = \frac{(1.645)^2 * 0.3 * (1 - 0.3)}{(0.03)^2}\).

Calculating the sample size

Carrying out the calculation gives a result of approximately \(844.37\). However, since we can't have a fraction of a participant in a sample, it's typical to round up to the next whole number, which will give us a sample size of \(845\).

Key concepts

Margin of Error

Understanding the margin of error is essential for anyone working with statistical data. It refers to the range in which the true population proportion is estimated to lie. For example, a margin of error of \(\pm 3\% \) suggests that the true value is within 3 percentage points above or below the reported statistic. In other words, if you're estimating a population proportion and report it as 30% with a margin of error of \(\pm 3\%\), the true proportion could be as low as 27% or as high as 33%. Calculating the appropriate sample size is crucial to ensure this precision. A smaller margin of error requires a larger sample size, keeping other factors constant, to reduce uncertainty in the estimation.

Confidence Level

The confidence level, often expressed as a percentage, indicates how certain we are that the true proportion falls within the margin of error. A \(90\%\) confidence level means that if the same population were sampled many times, approximately 90 out of 100 confidence intervals created from those samples would contain the true population proportion. It's a measure of reliability; the higher the confidence level, the more certain we are about our estimate. However, increasing the confidence level without changing the sample size would widen the margin of error. Therefore, selecting an appropriate confidence level is a balance between certainty and precision.

Population Proportion

The population proportion, denoted by \(p\), is a key factor in many statistical calculations. It represents the part of the population that has a particular attribute, expressed as a decimal between 0 and 1. For instance, if \(30\%\) of the population prefers a certain brand, \(p\) would be \(0.3\). Knowing the estimated population proportion is vital for accurate sample size calculation because it affects the variability in your data. Generally, a proportion close to \(0\) or \(1\) means less variability and may require a smaller sample size, whereas a proportion near \(0.5\) indicates more variability and typically requires a larger sample to obtain the same margin of error with the same confidence.

Z-score

In the context of sample size calculation, the Z-score corresponds to the selected confidence level and reflects the number of standard deviations a proportion is from the mean. For a \(90\%\) confidence level, the Z-score is \(1.645\). This value comes from a standard normal distribution table and reflects the point at which the area under the curve to the left is \(90\%\). A higher Z-score (related to a higher confidence level) results in a larger sample size needed to maintain the same margin of error. The Z-score is a critical component in the formula and plays a direct role in determining the precision of your population proportion estimate.