91Ó°ÊÓ

When we talk about sample proportion, we're referring to the fraction of individuals in a sample that exhibit a particular trait or characteristic. In the context of the provided exercise, the characteristic of interest is the use of social networking sites by hiring managers to screen job applicants.

The sample proportion, denoted by \( \hat{p} \), is the number of individuals in the sample with the characteristic divided by the total number of individuals in the sample. In our case, with 1,200 hiring managers using social networking sites out of a sample of 2,667, the sample proportion \( \hat{p} \) equals \( \frac{1200}{2667} \), which simplifies to approximately 0.449. This figure serves as an estimate for the true population proportion.

A representative sample accurately reflects the population from which it's drawn, ensuring that any conclusion drawn from the sample can be generalized to the population. To achieve this, the sample must be randomly selected, and its composition should mirror the diversity of the population in terms of key characteristics.

In our exercise, the survey's respondents are hiring managers and human resource professionals, and we assume that they are a representative sample of all such individuals in the general population. This assumption is crucial because it backs the validity of using the sample proportion to estimate the population proportion, particularly when constructing a confidence interval for that population proportion.

Sample size, denoted by \( n \), is the number of observations or individuals in a sample. The size of the sample plays a significant role in determining the precision of our estimates and the reliability of our statistical inferences.

The chosen sample size in our problem was 2,667, a figure which we assessed using the criteria that \( n\hat{p} \) and \( n(1 - \hat{p}) \) both exceed 5. This rule of thumb ensures that the Central Limit Theorem holds, and our sample proportion's distribution approaches normalcy, allowing us to confidently create a confidence interval for the population proportion. Our sample size of 2,667 satisfies this condition, with both \( 2667\times0.449 \) and \( 2667\times(1 - 0.449) \) being greater than 5.

The term 'binary population' refers to a scenario where each member of the population can be categorized into one of two groups based on a characteristic. This duality is essential when determining proportions since the characteristic is present in some members and absent in others.

In our exercise, the binary groups are hiring managers who do use social networking sites to screen job applicants and those who do not. This clear-cut separation allows for the calculation of the population proportion, which is the foundation for constructing a confidence interval. The binary nature of the population validates the methods used and the interpretability of the results.