Problem 76 - An Introduction to Mathematical Statistics and Its Applications

Question

Records show that 642 new students have just entered a certain Florida school district. Of those 642 , a total of 125 are not adequately vaccinated. The district's physician has scheduled a day for students to receive whatever shots they might need. On any given day, though, \(12 \%\) of the district's students are likely to be absent. How many new students, then, can be expected to remain inadequately vaccinated?

Step-by-Step Solution

Answer

Approximately 15 new students are expected to remain inadequately vaccinated.

Step 1 - Find the number of students likely to be absent

12% of the total number of new students in the district are predicted to be absent on any given day. To find this number, multiply the total number of new students in the district (642) with this percentage value. Hence, \(0.12 \times 642 = 77.04\). As we can't have a fraction of a student, this value should be rounded to the closest whole number which is 77.

Step 2 - Find the number of inadequately vaccinated students among the absentees

The problem is asking for the expected number of inadequately vaccinated new students remaining, which means those who are absent on the day of vaccination. The percentage of inadequately vaccinated students among the total students is \( \frac{125}{642} \approx 0.1947 \) or 19.47%. So, multiply this percentage with the number of students predicted to be absent to find out the number of inadequately vaccinated students among them. This is \(0.1947 \times 77 = 15.01\). Rounding to the closest whole number, it is 15.

Step 3 - Conclude the result

So, 15 students who are inadequately vaccinated are expected to be absent on the day of vaccination. Hence, these students can be expected to remain inadequately vaccinated.

Key Concepts

Vaccination Statistics

Vaccination statistics are an essential tool in public health for both understanding current health levels and planning future interventions. For this exercise, we're analyzing the data on student vaccinations in a specific school district. We know the exact number of students, which is 642, entering the district, and we also know that 125 of them are not adequately vaccinated. This data helps us create a base probability of finding students who need vaccinations: approximately 19.47% of the student population.

Understanding these statistics allows public health officials to pinpoint necessary interventions, like scheduling vaccinations, to improve health outcomes in the community. By using such data, schools can efficiently plan vaccination drives ensuring everyone gets vaccinated in a timely manner. This practical application of statistics emphasizes the need for accurate record-keeping and analysis, which are crucial in managing public health issues.

Student Absenteeism

Student absenteeism is a major concern in the education system. It's important to understand its impact, especially when planning important events like vaccination days. In this scenario, on any given day, 12% of students may be absent. This percentage comes from historical data and patterns observed in the district's attendance records.

This percentage helps us predict the number of students likely to miss out on school activities. In our case, out of 642 students, approximately 77 students may be absent. This absence affects not only their educational experience but also vital health services, such as vaccinations. It's crucial for schools to strategize accordingly, perhaps by providing alternative vaccination days or follow-ups, to cover those who might miss the initial drive. Effective planning considering student absenteeism ensures better participation in school health activities, safeguarding overall student health.

Mathematical Applications in Education

Mathematics plays a pivotal role in educational planning and problem-solving. This problem is an excellent example of using mathematical concepts to address real-world issues. Through percentage calculations, we can predict student behavior, such as absenteeism, and manage logistical aspects of health programs, like vaccinations.

We start by calculating the percentage of inadequately vaccinated students and then apply it to predict outcomes for absentee students. Using percentages and multiplication, educators can gather insights needed for decision-making. For instance, with a 12% absenteeism rate, we calculate the expected number of inadequately vaccinated students who might be absent on vaccination day—determined by multiplying 19.47% of the absentees, resulting in an estimated number of 15 students.

Such calculations not only aid in efficient resource allocation but also help in preparing contingency plans, ensuring minimal disruption in the implementation of educational health programs.