Question

Consider the following statement: A sample of size 100 was selected from those admitted to a particular college in fall 2012. The proportion of these 100 who were transfer students is \(\mathbf{0} . \mathbf{3 8}\). a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, \(p=0.38\) or \(p=0.38 ?\)

Short answer

The number in bold is a sample proportion and the correct notation to represent it is \( \hat{p} = 0.38 \).

Step-by-step solution

Identifying the proportion

The problem statement indicates that the number in bold, 0.38, is derived from a group of 100 students that were selected from a larger group who were admitted to a particular college in fall 2012. Since this data is from a subset of the larger group and not the whole population, it is a sample proportion.

Understanding the notation

In statistics, different notations are used for sample proportion and population proportion. The letter 'p' without hat (^) is used to represent population proportions. On the other hand, 'p' with a hat, also represented as \( \hat{p} \), is used to denote sample proportions.

Specifying the correct notation

From steps 1 and 2, it is evident that 0.38 is a sample proportion. So the correct notation is \( \hat{p} = 0.38 \). Inside \( \), notation should be used for any mathematical representation.

Key concepts

Population Proportion

When examining statistics, understanding the difference between a population proportion and a sample proportion is vital. The population proportion refers to the ratio of members in an entire group who have a particular attribute to the total number of members in that group.

For instance, if we are looking at a college's entire student body to find the proportion of transfer students, and 30% of all the students are transfers, then the population proportion, often denoted as 'p', is 0.30.

This value is a fixed number and represents a parameter of the population, but since it's often impractical or impossible to survey an entire population, researchers use a sample to estimate this value. In our exercise, the number 0.38 was calculated from a sample of 100 students, offering a glimpse into the likely characteristics of the larger population without needing to survey every single student.

Statistical Notation

Statistical notation is a symbolic method to represent different statistical values and parameters precisely and concisely. These notations are standardized across the field of statistics, allowing for clear communication and understanding between statisticians and students alike.

In the context of our exercise, the correct notation for a sample proportion is denoted by \( \hat{p} \) rather than 'p'. The hat symbol (^) indicates that this proportion is an estimate derived from a sample. Such fine distinctions in notation are crucial, as they differentiate between estimates and actual population parameters, and help avoid confusion when interpreting statistical data.

Therefore, for the exercise provided, the appropriate notation to express the sample proportion of 0.38 is \( \hat{p} = 0.38 \), reflecting the nature of the data as an estimation from a sample.

Statistics Education

Statistics education plays an essential role in enabling students to interpret and analyze data effectively. It provides the tools to understand variability, make informed decisions based on data, and to communicate findings accurately.

An important part of statistics education is teaching how to distinguish between different types of data, such as population versus sample data, and to use the correct methods and notation for each. Exercises like the one provided reinforce this understanding by asking students to apply the correct notation and reinforce methodological concepts.

Moreover, it is often useful to ground statistical education in real-world examples, providing context that can make the abstract concepts more tangible. This approach helps in better retention of the knowledge and its practical application. The step-by-step solutions offered in textbook resources facilitate a deeper grasp of these complex ideas, guiding the students through the process, and ensuring they understand not only the 'how' but also the 'why' behind the procedures.