Question

A building contractor has a chance to buy an odd lot of 5,000 used bricks at an auction. She is interested in determining the proportion of bricks in the lot that are cracked and therefore unusable for her current project, but she does not have enough time to inspect all 5,000 bricks. Instead, she checks 100 bricks to determine whether each is cracked. Describe the population and sample for this problem.

Short answer

In this scenario, the population is the entire lot of 5,000 bricks up for auction, and the sample is the 100 bricks that the contractor inspects.

Step-by-step solution

Identification of the Population

The population of a statistical study is the complete set of observations or data points that can be made. In this scenario, the population is the entire lot of 5,000 bricks that the contractor has a chance to buy at the auction. This is because the contractor is interested in the entire lot of bricks.

Identification of the Sample

A sample, on the other hand, is a subset of the population that is selected for the study. In this case, the sample would be the 100 bricks that the building contractor inspects to make a determination about the quality of the whole lot of bricks.

Key concepts

Population and Sample

In statistical sampling, distinguishing between a population and a sample is a fundamental step. The population refers to the entire group of objects or individuals we want to study or about which we want to make conclusions. For instance, when a building contractor wants to assess the quality of bricks in a lot, the entire set of 5,000 bricks in that lot represents the population. It's often impractical to examine every single member of a population due to constraints of time, resources, or accessibility.

A sample is a manageable portion of the population that is selected for examination. By analyzing this subset, reliable conclusions about the whole population can be drawn. In our example, the 100 bricks inspected by the contractor serve as the sample. This approach saves time and money, while still providing valuable insights into the quality of the entire lot. It's essential, however, to ensure the sample is representative of the population, which means the selected bricks need to reflect the variety and conditions of all the bricks in the lot.

Proportion Estimation

Proportion estimation is a statistical technique used to draw conclusions about the frequency of a particular attribute within a population. It's expressed as a percentage or a fraction. For the building contractor assessing brick quality, proportion estimation involves estimating the percentage of unusable or cracked bricks within the total lot of 5,000.

To carry this out, the proportion of defective bricks within the sample is calculated first. If out of 100 inspected bricks 10 are cracked, the sample proportion is \( \frac{10}{100} = 0.10 \) or 10%. This sample proportion serves as an estimate of the proportion of defective bricks in the entire lot. Accurate estimation helps in decision-making processes such as purchasing materials. It's important to recognize that the closer the sample reflects the actual population, the more reliable the proportion estimation will be.

Statistical Study

Conducting a statistical study involves systematic collection, review, and analysis of data to understand and draw conclusions about a particular subject or population. In every study, clearly defining the research question is crucial. For our building contractor, the research question is: What proportion of the 5,000 bricks are cracked and unsuitable for use?

The process entails several steps, including defining the population and sample, collecting data accurately from the sample, and then analyzing this data to estimate characteristics of the population. The conclusion drawn from this sample is then generalized to the population, which is why the sample's representativeness is paramount. Through statistical studies, we can make informed decisions without needing to assess every single instance within a population.

Sampling Techniques

The method by which the sample is selected from the population is critical for obtaining reliable results in statistical studies. Different sampling techniques serve different purposes, and their selection depends on the characteristics of the population and the goals of the study. Common techniques include:

• Simple Random Sampling: Every member of the population has an equal chance of being included in the sample. This can be done using random number tables or computer-generated random numbers.
• Stratified Sampling: The population is divided into subgroups, or strata, that share similar characteristics, and a sample is taken from each stratum.
• Cluster Sampling: The population is divided into clusters, often based on geography or another attribute, and entire clusters are randomly selected to be included in the sample.
• Systematic Sampling: A sample is drawn by selecting members of a population at regular intervals, such as every 10th brick in a stack.

In the example with the building contractor, if the bricks are homogeneously mixed, simple random sampling may suffice. The goal is to end up with a sample that accurately reflects the population, enabling a meaningful proportion estimation to be made.