Question

The principal at a large high school in a major city believes that the math teachers for each class assign drastically different amounts of homework to their respective students. Which method of sampling would be best to estimate the average number of hours of homework assigned to students by the different math teachers? A) Select one math teacher at random and survey the number of assigned homework hours in each of his or her classes. B) Post an online poll to the school's social media account that asks students how much math homework they get each night. C) Select 5 students from each math class in the high school and then record the number of hours assigned to each student selected. D) Select the student with the highest grade in each math class and then survey those students to see how much homework they do each night.

Short answer

The best sampling method to estimate the average number of hours of homework assigned to students by different math teachers is \(Option \thinspace C:\) selecting \(5\) students from each math class in the high school and then recording the number of hours assigned to each student selected. This method provides the most representative and unbiased sample data.

Step-by-step solution

Option A Evaluation

Selecting a single math teacher at random and surveying the number of assigned homework hours in each of their classes would not be a good representation of the entire school's math teachers. The resulting data would only represent the assignments given by one teacher, potentially leading to biased and unrepresentative results.

Option B Evaluation

Posting an online poll to the school's social media account would make the sample self-selecting, and it would likely suffer from a low response rate. As a result, there is no guarantee that it would accurately represent the student body or the homework given by all math teachers in the high school. This method could result in a response bias as students who might have a certain opinion about the matter are more likely to participate.

Option C Evaluation

Selecting five students from each math class in the high school and then recording the number of hours assigned to each student selected offers a more systematic approach. By selecting students from each class, we ensure that the sample is stratified according to classes and teachers and more likely to be representative of the student body. This method seems to be more effective in estimating the average number of hours of homework assigned to students by different math teachers.

Option D Evaluation

Selecting the student with the highest grade in each math class could introduce bias into the sample, as students with high grades could have different homework assignments or habits than their peers. This method will not provide an accurate representation of the homework assigned by the different math teachers across the entire student population.

Concluding Decision

Based on the evaluations above, the best sampling method for estimating the average number of hours of homework assigned to students by different math teachers is Option C: "Select 5 students from each math class in the high school and then record the number of hours assigned to each student selected." This method offers the most representative and unbiased sample data of the options provided.

Key concepts

Stratified Sampling

Stratified sampling is a technique used to ensure that subgroups within a population are adequately represented within the sample. Imagine dividing the entire population into distinct layers or 'strata' based on shared characteristics. These strata are then sampled individually, often in proportion to their size in the overall population. This method is particularly effective when researchers believe that the different strata exhibit distinct behaviors or characteristics relevant to the study.

In the context of the SAT math sampling exercise, the principal wants to estimate the average number of homework hours assigned by different math teachers in a high school. Stratified sampling would be the ideal method because it would involve taking a sample from each math class — every class acting as a 'stratum.' By doing this, the principal can ensure that each teacher’s homework assignment habits are accounted for in the study. This approach minimizes the risk of sampling bias and provides a more precise estimate of the average homework load. It is more representative because it acknowledges the diversity inherent in the population of math classes and teachers.

Sampling Bias

Sampling bias occurs when the sample collected does not accurately represent the population from which it was drawn. It can significantly affect the validity of research findings. Biases can emerge due to non-random selection of participants, underrepresentation of particular groups, or overrepresentation of others. If not accounted for, such biases can lead to conclusions that are not genuinely reflective of the entire population.

In our example, sampling bias could be introduced if only one math teacher's students (Option A) or only the highest-grade students (Option D) were surveyed. These samples would be skewed because they do not encompass the diversity of the entire student body and its varying experiences with homework. Such a biased sample would produce results that favor one segment's characteristics, thereby providing inaccurate insights into the homework trends set by all math teachers. Being aware of and avoiding sampling bias is crucial for researchers who aim to draw conclusions that are applicable to the whole population they are studying.

Representative Sample

A representative sample captures the actual characteristics and diversity of the population as closely as possible. The goal is to create a 'miniature' of the population that is true to its varying elements. This allows for findings and conclusions drawn from the sample to be reasonably generalized to the entire population. Representativeness is essential to ensure that all subsets within the population have a fair chance of being included.

Using Option C, where five students from each math class are selected, would likely create a representative sample of all students in the high school. This is because it accounts for the possible differences in homework amounts across different classes and teachers. The strategy aims to reflect the broader homework trends and habits seen within the school and is therefore integral to producing a reliable estimate of the average homework hours. In studies like the one presented, ensuring representativeness is key to making informed decisions based on the data.