Question

A researcher wants to estimate the proportion of students enrolled at a university who are registered to vote. Would the standard error of the sample proportion \(\hat{p}\) be larger if the actual population proportion was \(p=0.4\) or \(p=0.8\) ?

Short answer

The standard error of the sample proportion, \(\hat{p}\), would be larger if the actual population proportion was \(p=0.4\), compared to if it was \(p=0.8\).

Step-by-step solution

Understand the formula for calculating the standard error of a proportion

The formula used to calculate the standard error of a proportion is \(\sqrt{p(1-p)/n}\), where p is the population proportion and n is the sample size. This formula shows us that the standard error of a proportion is affected by the population proportion and the size of the sample.

Substitute the given values into the formula

First, substitute \(p=0.4\) into the formula. This will produce \(\sqrt{0.4(1-0.4)/n} = \sqrt{0.24/n}\). Now, substitute \(p=0.8\) into the formula. This will produce \(\sqrt{0.8(1-0.8)/n} = \sqrt{0.16/n}\).

Conclusion

Since we are not given the exact number of samples (n), we can still compare the two scenarios. Looking at the results, it is clear that the standard error would be larger if \(p=0.4\) than if \(p=0.8\). This is because the product of \(p(1-p)\) is larger when \(p=0.4\) than when \(p=0.8\).

Key concepts

Population Proportion

When researching statistical data, it's crucial to understand the concept of population proportion, often denoted as 'p'. It represents the fraction of members in a population that exhibit a particular attribute. For example, in a study of voters, 'p' could be the proportion of all students at a university who are registered to vote.

The value of 'p' plays a significant role in calculating various statistics, including the standard error of a proportion. It's interesting to note that as 'p' moves away from the extremes of 0 and 1, the product 'p(1-p)', which forms part of the standard error formula, increases. Consequently, the standard error is larger in scenarios where 'p' is closer to 0.5, since this is where the product 'p(1-p)' reaches its maximum value.

Sample Size

The term 'sample size', denoted as 'n', refers to the number of observations or measurements taken from a population to form a sample. This is a crucial concept because the size of the sample directly impacts the precision of statistical estimates.

A larger sample size generally leads to more precise estimates of population parameters, as it more closely reflects the true distribution of the population. This is why researchers aim for a larger 'n' in their studies. Within the standard error formula, \(\sqrt{p(1-p)/n}\), we see that the standard error is inversely proportional to the square root of the sample size. This means that as 'n' increases, the standard error decreases, leading to more accurate estimates of the population proportion.

Sampling Distribution

A sampling distribution is a probability distribution of a statistic that is formed by considering all possible samples of a given size from a population. It is a fundamental concept in statistics, as it allows us to understand the behavior of sample estimates and to make inferences about the population.

The standard error of a proportion is a measure of how much we expect the sample proportion (\(\hat{p}\)) to vary from the true population proportion ('p'). This variability is depicted in the sampling distribution of \(\hat{p}\). When the sampling distribution of \(\hat{p}\) is centered around 'p' with a smaller standard error, the spread is narrow, indicating that samples are likely to yield estimates close to the population proportion. Conversely, a larger standard error indicates a wider spread, suggesting a higher chance of obtaining sample proportions that differ from 'p'. Understanding this distribution helps researchers gauge the reliability of their sample estimates and the likelihood of drawing accurate conclusions about the population.