The Table Below Shows The Probability Distribution Of Student Ages In A High School With 1500 Students. What Is The Expected Value For The Age Of A Randomly Chosen Student?$\[ \begin{tabular}{|c|c|c|c|c|c|c|} \hline Age & 13 & 14 & 15 & 16 & 17 &
Introduction
In probability theory, the expected value is a measure of the central tendency of a probability distribution. It represents the long-run average value that a random variable is expected to take on. In this article, we will calculate the expected value of the age of a randomly chosen student in a high school with 1500 students.
Probability Distribution of Student Ages
The table below shows the probability distribution of student ages in a high school with 1500 students.
| Age | Frequency | Probability |
|---|---|---|
| 13 | 150 | 0.1 |
| 14 | 300 | 0.2 |
| 15 | 400 | 0.27 |
| 16 | 300 | 0.2 |
| 17 | 150 | 0.1 |
Calculating the Expected Value
To calculate the expected value, we need to multiply each age by its probability and sum up the results.
Let's denote the age of a randomly chosen student as X. The expected value of X, denoted as E(X), is given by:
E(X) = (13 × 0.1) + (14 × 0.2) + (15 × 0.27) + (16 × 0.2) + (17 × 0.1)
Step-by-Step Calculation
Now, let's perform the step-by-step calculation of the expected value.
- Multiply each age by its probability:
(13 × 0.1) = 1.3 (14 × 0.2) = 2.8 (15 × 0.27) = 4.05 (16 × 0.2) = 3.2 (17 × 0.1) = 1.7
- Sum up the results:
E(X) = 1.3 + 2.8 + 4.05 + 3.2 + 1.7 = 13.15
Conclusion
The expected value of the age of a randomly chosen student in a high school with 1500 students is 13.15 years. This means that if we were to randomly select a student from the school, we would expect their age to be around 13.15 years.
Interpretation of Results
The expected value of 13.15 years indicates that the majority of students in the school are likely to be between 13 and 14 years old. This is because the probability distribution is skewed towards the lower ages, with a higher frequency of students in the 13-14 age range.
Limitations of the Study
One limitation of this study is that it assumes that the probability distribution of student ages is stationary, meaning that it does not change over time. In reality, the probability distribution of student ages may change due to various factors such as demographic changes, migration, or changes in birth rates.
Future Research Directions
Future research could explore the following directions:
- Longitudinal study: Conduct a longitudinal study to examine how the probability distribution of student ages changes over time.
- Comparison with other schools: Compare the probability distribution of student ages in this school with other schools to identify any differences or similarities.
- Analysis of demographic factors: Analyze the impact of demographic factors such as socioeconomic status, ethnicity, or geographic location on the probability distribution of student ages.
Q: What is the expected value of student ages in a high school?
A: The expected value of student ages in a high school is 13.15 years, based on the probability distribution of student ages in a school with 1500 students.
Q: How is the expected value calculated?
A: The expected value is calculated by multiplying each age by its probability and summing up the results. In this case, the expected value is calculated as:
E(X) = (13 × 0.1) + (14 × 0.2) + (15 × 0.27) + (16 × 0.2) + (17 × 0.1)
Q: What does the expected value represent?
A: The expected value represents the long-run average value that a random variable is expected to take on. In this case, it represents the average age of a randomly chosen student in the high school.
Q: Why is the expected value important?
A: The expected value is important because it provides a measure of the central tendency of a probability distribution. It can be used to make predictions about the behavior of a random variable and to make informed decisions.
Q: Can the expected value be used to make predictions about individual students?
A: No, the expected value is a population-level measure and cannot be used to make predictions about individual students. Each student's age is a random variable, and the expected value only provides a general idea of the average age of the population.
Q: How does the expected value change over time?
A: The expected value can change over time due to changes in the probability distribution of student ages. For example, if the school experiences an increase in the number of students in the 16-17 age range, the expected value would increase accordingly.
Q: Can the expected value be used to compare different schools?
A: Yes, the expected value can be used to compare different schools. By calculating the expected value of student ages for each school, you can compare the average age of students in each school and identify any differences or similarities.
Q: What are some limitations of the expected value?
A: Some limitations of the expected value include:
- It assumes that the probability distribution of student ages is stationary, meaning that it does not change over time.
- It does not take into account individual differences between students.
- It is a population-level measure and cannot be used to make predictions about individual students.
Q: What are some future research directions for the expected value of student ages?
A: Some future research directions for the expected value of student ages include:
- Conducting a longitudinal study to examine how the probability distribution of student ages changes over time.
- Comparing the probability distribution of student ages in this school with other schools to identify any differences or similarities.
- Analyzing the impact of demographic factors such as socioeconomic status, ethnicity, or geographic location on the probability distribution of student ages.
By exploring these research directions, we can gain a deeper understanding of the expected value of student ages and its implications for education policy and practice.