Question

A random sample of 100 employees of a large company included 37 who had worked for the company for more than one year. For this sample, \(\hat{p}=\frac{37}{100}=0.37\). If a different random sample of 100 employees were selected, would you expect that \(\hat{p}\) for that sample would also be \(0.37 ?\) Explain why or why not.

Short answer

No, we would not necessarily expect the sample proportion \(\hat{p}\) to be exactly 0.37 in another random sample of 100 employees, due to the inherent randomness and variability in statistical sampling.

Step-by-step solution

Understand sample proportions

In statistics, a sample is a subset of a larger population. The sample proportion \(\hat{p}\) is the ratio of the number of successful outcomes to the total number of trials in the sample. We calculated \(\hat{p}\) for the first survey to be 0.37.

Understand sample variability

Variability is at the heart of statistics. Even if the parameters of the population don't change, every new random sample can lead to different statistics. This is due to the chance variation that, in general, different random samples from the same population will not be identical.

Answer the question

So if we were to take another sample of 100 employees, it is not expected for the sample proportion \(\hat{p}\) to be exactly 0.37 because of the inherent randomness of sampling. While the proportion for the new sample is likely to be somewhat close to 0.37, especially if the sample size is large and the sampling is random, it's not expected to be exactly the same due to sample variability in statistics.

Key concepts

Sample Proportion

Understanding the concept of sample proportion is fundamental when dealing with statistics, especially in the context of gauging certain characteristics within a population. In the exercise concerning company employees, the sample proportion, denoted as \(\hat{p}\), represents the fraction of employees who have been with the company for over a year. This measure is calculated by dividing the number of individuals who meet the criterion (37 employees) by the total number of individuals in the sample (100 employees), resulting in a proportion of 0.37.

The sample proportion provides us with an estimate of the true proportion of the entire population that would satisfy the same criterion. However, it is crucial to recognize that \(\hat{p}\) varies from sample to sample due to sampling variability, a concept we will explore in more detail. It's important for students to grasp the idea that sample proportions are merely snapshots of a larger picture, and should they conduct a similar study with another random sample, they might find slightly different proportions.

Random Sampling

Random sampling is a cornerstone concept in statistics as it allows for the collection of unbiased data from a population. When a random sample is collected, every member of the population has an equal opportunity of being selected. This unbiased selection process is imperative because it ensures that the sample represents the population well, minimizing skewed results that could occur from a non-random sampling method.

In the provided exercise, a random sample of 100 employees gives us a measure of who has worked for the company for more than one year. If the process were truly random, this would imply that every employee in the large company had an equal chance of being chosen for the sample. Therefore, the resulting proportion would theoretically reflect the true proportion of the overall employee population that satisfies the criterion. However, even with a random sampling method, the exact sample proportion can differ each time because each subset of the population has the chance to yield different outcomes. Understanding this helps students appreciate why replication of the sampling process could lead to variance in results, which is normal and expected.

Statistical Variability

Statistical variability, sometimes known as statistical variation, refers to the natural differences that occur when different samples are taken from the same population. This variation is due to the randomness inherent in the sampling process. Given that each sample is just a small representation of the entire population, some fluctuation in sample statistics like the sample proportion would occur.

As seen in the exercise, the sample proportion of the first sample of employees is 0.37. If we were to take a second random sample of 100 employees from the same population, the sample proportion \(\hat{p}\) would almost certainly differ from 0.37. This is not an indication of error but rather a reflection of natural variability within the sampling process. The degree of variability can be influenced by the size of the sample - larger samples tend to have less variability. It's essential for students to recognize that such variability is not only normal but expected, and it reflects the true nature of dealing with random variables in statistics.