Question

Consider taking a random sample of size 10 from the population of students at a certain college and asking each of the 10 students whether or not they smoke. In the context of this setting, explain what is meant by the sampling distribution of \(\hat{P},\) the ordinary sample proportion.

Short answer

The sampling distribution of \(\hat{P}\) is the distribution of sample proportions that would be obtained from taking many random samples of size 10 from the student population at the college to estimate the proportion of students who smoke.

Step-by-step solution

Understanding the Concept of Sampling Distribution

The sampling distribution of a statistic is the distribution of that statistic, considered as a random variable, when derived from a large number of samples of the same size. It tells us how the statistic varies from sample to sample.

Defining the Ordinary Sample Proportion

In this context, the ordinary sample proportion, denoted as \(\hat{P}\), is a statistic that estimates the proportion of the student population at the college who smoke. It is calculated by dividing the number of students in the sample who smoke by the total number of students in the sample.

Describing the Sampling Distribution of \(\hat{P}\)

The sampling distribution of \(\hat{P}\) refers to the distribution of the sample proportions \(\hat{P}\) that we would obtain if we took many samples of size 10 from the student population. It provides information about the variability of \(\hat{P}\), its expected value, and its standard deviation.

Key concepts

Sample Proportion

When conducting studies or surveys, researchers often use a small, selected group from a larger population to make inferences about the whole. The sample proportion is a measure that gives us insight into the expected characteristic of the larger group. It is represented by the symbol \(\hat{P}\) and is calculated by dividing the number of outcomes of interest in the sample by the total number of observations in the sample. For example, if you want to know the proportion of students who smoke at a certain college, you would randomly choose a group of students (the sample), ask them if they smoke, and use the ratio of smokers to the total sampled students as your sample proportion.

In the exercise, a random sample of size 10 is taken from the student population at a college to determine the smoking rate, \(\hat{P}\). If, say, 3 out of the 10 students sampled are smokers, the sample proportion \(\hat{P}\) would be \(\frac{3}{10}=0.3\). This means that, based on our sample, we estimate that 30% of the college's students are smokers. It's important to remember that this is an estimation based on the sample and will likely vary if the sample size or the sample itself changes, due to statistical variability.

Statistical Variability

One of the fundamental concepts in statistics is statistical variability, which refers to the extent to which data points in a set differ from each other and from the average of the data. In the context of sample proportions, statistical variability describes how much the sample proportion \(\hat{P}\) would vary from sample to sample. It is influenced by many factors, including the size of the sample and the inherent variability in the population.

To measure the statistical variability of \(\hat{P}\), standard deviation is often used. It quantifies the dispersion of the sample proportions around the expected value (mean) of \(\hat{P}\) in the sampling distribution. A larger standard deviation indicates more variability amongst the sample proportions. In our smoking example, if additional samples of 10 students yield proportions like 0.2, 0.3, and 0.4, we can see there is some variability in our sample proportions. A larger sample size usually results in a smaller standard deviation, indicating that the sample proportion will more closely estimate the population proportion with more precision.

Population Parameter Estimation

The main goal of using sample data is to make inferences about a population parameter. Population parameter estimation involves using statistics, such as the sample proportion, to estimate values of population parameters like the true proportion of smokers in a college. This estimation comes from the sampling distribution, which is the distribution of the statistic across many samples.

In practice, we rarely know the true population proportion, so we use the sample proportion as an estimate. Consequently, the reliability of our estimation depends on how well our sample represents the population. To improve this estimation, statisticians often use larger sample sizes and ensure that the sampling method minimizes bias. The precision of our estimate can be quantified using confidence intervals, which give a range of plausible values for the population parameter. For instance, we might be 95% confident that the proportion of college students who smoke falls between 20% and 40%, based on the variability observed in our samples' proportions and other relevant calculations. Providing students with the knowledge of how to estimate these intervals helps them understand the level of certainty associated with the sample proportion as an estimator for the population proportion.