Question

A random sample of 50 registered voters in a particular city included 32 who favored using city funds for the construction of a new recreational facility. For this sample, \(\hat{p}=\frac{32}{50}=\) 0.64 . If a second random sample of 50 registered voters was selected, would it surprise you if \(\hat{p}\) for that sample was not equal to 0.64 ? Why or why not?

Short answer

No, it would not be surprising if \(\hat{p}\) for the second sample was not equal to 0.64 due to sample variability, which is inherent to any random sampling method. Despite the sample size and process being the same, the sample proportion can vary from sample to sample.

Step-by-step solution

Understanding the concept of sample proportion

The sample proportion (\(\hat{p}\)) is defined as the ratio of the number of successes to the sample size. In this exercise, it is given by \(\hat{p} = \frac{32}{50}\), which equals 0.64. This means that 64% of the sampled voters favor using city funds for the construction of a new facility.

Understanding variability in sampling

Even though an identical sample procedure is used, the proportion in a second sample will not necessarily be the same as in the first sample. This is due to the variability inherent in random sampling. This is why the value of \(\hat{p}\) can differ from one sample to another even if the sizes are the same. One important thing to remember is that larger samples tend to have less variability.

Determining if it would be surprising to get a different \(\hat{p}\) for the second sample

Given the understanding that \(\hat{p}\) varies between samples, it would not be surprising to get a different \(\hat{p}\) in a second sample. Therefore, even if a second random sample of 50 registered voters were selected, it would not be surprising if \(\hat{p}\) was not equal to 0.64.

Key concepts

Random Sampling

When studying statistics, one fundamental process is random sampling, a technique used to select a group of subjects (a sample) for study from a larger group (a population). Each individual is chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process. This method ensures that the sample represents the population as a whole without bias.

Random sampling is essential for the integrity of statistical analysis because it mitigates the influence of external variables. If a sample isn't randomly selected, the results might not accurately reflect the broader population, leading to false conclusions. In educational contexts such as textbook problem-solving, we often encounter hypothetical scenarios where random sampling is used to illustrate statistical concepts, including the calculation of sample proportions like \( \hat{p} \).

Variability in Sampling

A crucial point for students to understand is the concept of variability in sampling. This refers to the expected differences in sample statistics from one sample to another. The sample proportion \( \hat{p} \) is a statistic that estimates a population proportion, and because of random sampling, different samples will typically yield different results.

Understanding that variability exists and is a natural part of the random sampling process helps explain why the proportions from two different samples of the same population might not match exactly. Explaining this concept to students with real-life examples or simulations can significantly clarify the nature of sampling variability and its impact on statistical conclusions.

To emphasize this concept, let's consider that if another random sample of the same size is taken from the population, the proportion favoring the use of city funds might slightly differ due to pure chance.

Sample Size

Another integral statistical concept is sample size. The sample size, denoted as \( n \), is the number of observations in a sample and has a direct impact on the precision of the estimated population parameter, in this case, the proportion \( \hat{p} \). In general, a larger sample size can reduce uncertainty in the estimate and provides more accurate and reliable results.

However, there is a balance to strike, as larger samples require more resources and can be more time-consuming to collect. Statisticians use various methods to determine the optimal sample size needed for sufficient precision without unnecessary expenditure. When students are encouraged to reflect on sample sizes in problem-solving exercises, they gain a deeper understanding of the balance between the efficiency and reliability of statistical results.