Question

A random sample of 1,000 students at a large college included 428 who had one or more credit cards. For this sample, \(\hat{p}=\frac{428}{1,000}=0.428 .\) If another random sample of 1,000 students from this university were selected, would you expect that \(\hat{p}\) for that sample would also be 0.428 ? Explain why or why not.

Short answer

No, we would not necessarily expect the proportion \(\hat{p}\) to be exactly 0.428 in a new sample, though the new \(\hat{p}\) would likely be close to 0.428. This variation is due to sampling variability and the nature of random sampling, where each sample is independent and hence may yield a slightly different \(\hat{p}\).

Step-by-step solution

Understanding the nature of random sampling

Random sampling is based on the principle that each sample drawn from the same population should be independent and not influenced by the previous samples. Therefore, a new sample of 1,000 students may or may not result in the proportion \(\hat{p}=0.428\). It could be slightly different due to the randomness of the selection.

Understanding sample proportions

The sample proportion \(\hat{p}=0.428\) is an estimate of the actual population proportion. While the same sample proportion could be obtained in a subsequent sample, it is not guaranteed. It could be slightly more or less than 0.428, depending on the sample selected.

Using probability theory

According to probability theory, the expected value is the same as the population parameters when dealing with proportions. Therefore, if the actual proportion of students with credit cards in the entire college is 0.428, then we expect that the proportion from another sample would be close to 0.428. However, due to sampling variability, we cannot say with absolute certainty that it would be exactly 0.428.

Key concepts

Sampling Variability

When we pull a random sample from a population, like selecting 1,000 students from a college to find out how many have credit cards, each sample can yield different results. This is a fundamental concept known as sampling variability. It refers to the natural fluctuations that occur in statistics, such as a sample proportion, from one sample to another.

These differences are due to the random nature of sampling. Each student has a certain likelihood of holding a credit card, but until they’re actually selected and observed, we can’t know for sure if they will increase or decrease the sample proportion of cardholders, \( \hat{p} \). In practice, this means that if we repeated the sample process multiple times, we'd likely see different values of \( \hat{p} \) each time, even if the underlying proportion in the population remains unchanged.

While we might hope to get the same result of 0.428 each time, sampling variability tells us that this will not always be the case. The variability is influenced by the sample size, the variance in the population, and the sampling method. For educational purposes, it's critical to grasp that \( \hat{p} \) is an estimate of the true population proportion and it's normal for it to vary from sample to sample.

Sample Proportion

The sample proportion, denoted as \( \hat{p} \), is a statistic that estimates the proportion of individuals in a population who have a certain characteristic, based on a sample. In the context of our example, where we found that 428 out of 1,000 randomly sampled students have credit cards, our sample proportion is 0.428.

This number is a snapshot of the larger student body, aimed at giving us a sense of how common credit card ownership is among all students. It's important to understand that the sample proportion is just that—an estimate. It can vary from the true proportion due to the randomness inherent in sampling.

With this in mind, improving our understanding of the sample proportion involves acknowledging that it's a reflection of our specific sample at a specific time. A different random sample could lead to a slightly higher or lower proportion due to the different individuals who make up that sample. Therefore, while \( \hat{p} \) can serve as a useful estimate of the actual proportion, it's essential to consider it within the context of sampling error and respect its limitations as an estimate.

Probability Theory

Probability theory provides the mathematical foundation for understanding random processes, such as random sampling. According to probability theory, we can calculate expectations and make predictions about phenomena like sample proportions.

In the case of our college students and their credit card ownership, probability theory would say that if the true proportion of students with credit cards in the entire college is around 0.428, and we sample students randomly, then on average, our sample proportion should be close to the true proportion. However, 'on average' does not mean 'always.' Each individual sample might still vary due to chance alone.

Expected Value and Variance

Two key concepts from probability theory that help us with this include the expected value, which is what we predict to happen on average, and variance, which tells us how much the results are likely to differ from the expected value. These ideas from probability theory are essential as they allow us to quantify the uncertainty of our sample proportion, \(\hat{p}\), and give us tools to assess the reliability of our estimates.