Question

A random sample will be selected from the population of all students enrolled at a large college. The sample proportion \(\hat{p}\) will be used to estimate \(p,\) the proportion of all students who use public transportation to travel to campus. For which of the following situations will the estimate tend to be closest to the actual value of \(p ?\) i. \(\quad n=300, p=0.3\) ii. \(\quad n=700, p=0.2\) iii. \(n=1,000, p=0.1\)

Short answer

The estimate from the third scenario (\(n=1,000, p=0.1\)) will tend to be closest to the actual value of \(p\). This is because it has the largest sample size, and larger sample sizes generally produce more accurate estimates.

Step-by-step solution

Understanding the concept

The accuracy of an estimate tends to increase with the sample size. As the sample size grows, the sample proportion \(\hat{p}\) tends to get closer to the population proportion \(p\). So, a larger sample size is expected to provide a more accurate estimate of \(p\).

Identifying the largest sample size

Among the given situations, the third scenario (\(n=1,000, p=0.1\)) has the largest sample size of 1000 students. Therefore, it is expected that this scenario will provide the most accurate estimate of the population proportion \(p\).

Choosing the best estimate

Comparing the sample sizes in all cases, the scenario with the largest sample size, which is the third scenario (\(n=1,000, p=0.1\)), is expected to provide an estimate of \(p\) that is closest to the actual value.

Key concepts

Sample Size

Understanding the implications of sample size in population proportion estimation is crucial for students and researchers alike. In statistics, we often want to make inferences about a larger population based on a sample—a smaller, manageable version of the population that is thought to be representative. The key to making accurate inferences is choosing an appropriate sample size. A larger sample size generally leads to more precise estimates of the population parameter because it reduces the impact of random sampling error.

For instance, if you’re trying to determine the percentage of college students who use public transportation, a sample size of 30 might give you a rough estimate, but a sample size of 1,000 will likely provide a much clearer picture. Larger sample sizes reduce variability and increase the likelihood that the sample proportion will mirror the true population proportion, enhancing the reliability of your study.

Sample Proportion

The concept of sample proportion, denoted as \(\hat{p}\), represents the estimated percentage of individuals in a sample that have a certain characteristic. When conducting surveys or studies, researchers calculate the sample proportion to draw conclusions about the corresponding population proportion \(p\). It is derived by dividing the number of individuals with the characteristic in the sample by the total sample size.

For example, in estimating how many students commute by public transport, researchers would divide the number of surveyed students who do so by the total number of surveyed students. This fraction is the sample proportion, which provides an estimate but might not be the exact population proportion due to sampling variation. That's why it's often accompanied by a confidence interval, which gives a range within which the true population proportion likely falls.

Accuracy of Estimates

The accuracy of estimates in statistical analysis is paramount. It dictates how closely an estimate—such as a sample proportion—approximates the true population proportion. Various factors impact the accuracy of these estimates, including sample size, sample variability, method of data collection, and the use of appropriate statistical techniques.

The Central Limit Theorem plays a significant role too, as it allows us to approximate the distribution of our estimate, thus influencing how we calculate confidence intervals around our estimate. In the context of the exercise, the larger the sample size, the smaller the margin of error, resulting in a closer estimate to the actual value. Accuracy is essential, especially when making policy or resource allocation decisions based on statistical data.

Statistics

Statistics as a branch of mathematics is integral to many fields—it allows us to collect, analyze, interpret, present, and organize data. It's the foundation upon which decisions are made in business, science, and many other areas. For students grappling with concepts like population proportion estimation, statistics offer the tools to make sense of data collected from studies and experiments.

Understanding fundamental statistical concepts, such as mean, median, standard deviation, and correlation, can empower students to conduct their own research and critically evaluate the validity of studies they encounter in academia and the real world. Moreover, the application of these concepts, such as choosing an appropriate sample size for a study or estimating a population proportion accurately, is where your statistics knowledge is put into practice.