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 \(\pm 5 \%\) with \(95 \%\) confidence.

Short answer

After computing the above equation, one finds that the minimum sample size needed to estimate the population proportion within the ±5% margin of error with 95% confidence is approximately 385.

Step-by-step solution

Identify Margin of Error

The margin of error presented is ±5%, which when converted to a decimal form is ±0.05.

Identify Confidence Level

The confidence level given is 95%. The Z-score of a 95% confidence level is 1.96.

Apply the Formula for Sample Size

The formula for determining the sample size needed for a given margin of error E and Z-score (Z) for a population proportion p, when we don't know the population standard deviation, is \(n = (Z^2 \cdot p \cdot ( 1-p )) / E^2\). However, since we don't know the population proportion p, we usually use the safe assumption of p=0.5 that maximizes the sample size, yielding \(n = (Z^2 \cdot 0.5 \cdot 0.5) / E^2\). Substituting the values into the sample size formula gives: \(n = (1.96^2 \cdot 0.5 \cdot 0.5) / 0.05^2\).

Key concepts

Margin of Error

Understanding the margin of error is critical when analyzing the results of a survey or poll. It represents the span of values above and below the survey proportion, within which the true population proportion is estimated to lie. In simpler terms, the margin of error accounts for the uncertainties present in any form of statistical sampling. For instance, if the margin of error were to be ±5%, and your survey outcome indicated a 60% favorability rating, the true population favorability could be as low as 55% or as high as 65%.

In our example exercise, a margin of error of ±5% implies that the researcher is willing to accept a 5% potential discrepancy between the estimated proportion and the true value within the population. Lower margins of error require larger sample sizes, which makes the study potentially more accurate but also more resource-intensive. The margin of error is dependent on the sample size, the confidence level, and the variability within the population — in this case expressed by the population proportion.

Confidence Level

The confidence level signifies the degree of certainty we have that our sample accurately reflects the population. The most common confidence levels used in research are 90%, 95%, and 99%. A 95% confidence level, used in the exercise, means that if the survey or experiment were to be repeated multiple times, the calculated confidence interval would contain the true population parameter 95 out of 100 times.

In relation to the margin of error, the higher the confidence level you desire, the larger the sample size you'll need. This is because a higher confidence level increases the range of the confidence interval. The confidence level corresponds to a Z-score in the standard normal distribution — the number of standard deviations from the mean to capture the desired confidence interval. For a 95% confidence level, the Z-score is 1.96, which defines how far from the observed sample proportion we expect the true population proportion to fall, with the given level of confidence.

Population Proportion

The population proportion is a measure that represents the fraction of the population that possesses a certain characteristic or attribute. In surveying and research, we usually aim to estimate this proportion through sampling, due to the impracticality of surveying an entire population. The symbol 'p' denotes this value in the sample size formula.

When the true population proportion is unknown, which is often the case, a value of 0.5 (or 50%) is used to calculate the sample size, as this assumes maximum variability and therefore maximizes the required sample size. This is the most conservative estimate and ensures the sample size is large enough to account for an unknown level of variability within the population. In the case of our exercise, adopting 0.5 for the population proportion simplifies the sample size formula and ensures that our sample is representative enough for a reliable estimation of the true proportion with the given margin of error and confidence level.