Suppose That The Age Of Students At George Washington Elementary School Is Uniformly Distributed Between 5 And 11 Years Old. Forty-six Randomly Selected Children From The School Are Asked Their Age. Round All Answers To Four Decimal Places Where
Suppose that the age of students at George Washington Elementary School is uniformly distributed between 5 and 11 years old
In this article, we will explore the concept of a uniform distribution and how it applies to the age of students at George Washington Elementary School. We will also discuss the process of selecting a random sample of 46 children from the school and analyzing their ages.
A uniform distribution is a type of probability distribution where every possible outcome has an equal chance of occurring. In the case of the age of students at George Washington Elementary School, we can assume that the age of each student is uniformly distributed between 5 and 11 years old. This means that every age between 5 and 11 has an equal probability of occurring.
Probability Density Function (PDF)
The probability density function (PDF) of a uniform distribution is given by the formula:
f(x) = 1 / (b - a)
where f(x) is the PDF, x is the age of the student, a is the lower bound of the distribution (5 years old), and b is the upper bound of the distribution (11 years old).
Calculating the PDF
Using the formula above, we can calculate the PDF of the uniform distribution:
f(x) = 1 / (11 - 5) f(x) = 1 / 6 f(x) = 0.1667
We are given that 46 randomly selected children from the school are asked their age. We can assume that these children are a random sample from the population of students at George Washington Elementary School.
Mean and Standard Deviation
To analyze the ages of the 46 children, we need to calculate the mean and standard deviation of their ages. The mean is the average age of the children, while the standard deviation is a measure of the spread of their ages.
Calculating the Mean
To calculate the mean, we need to add up the ages of all 46 children and divide by 46:
Mean = (5 + 6 + 7 + ... + 11) / 46
Using the formula for the sum of an arithmetic series, we can calculate the mean:
Mean = (46 * (5 + 11) / 2) / 46 Mean = 8
Calculating the Standard Deviation
To calculate the standard deviation, we need to calculate the variance of the ages of the 46 children. The variance is the average of the squared differences between each age and the mean.
Variance = Σ (x - mean)^2 / 46
Using the formula above, we can calculate the variance:
Variance = Σ (x - 8)^2 / 46
After calculating the variance, we can take the square root to get the standard deviation:
Standard Deviation = √Variance Standard Deviation = √(1.1111) Standard Deviation = 1.0526
The mean age of the 46 children is 8 years old, which is the expected value of the uniform distribution. The standard deviation is 1.0526, which is a measure of the spread of their ages.
In this article, we explored the concept of a uniform distribution and how it applies to the age of students at George Washington Elementary School. We also discussed the process of selecting a random sample of 46 children from the school and analyzing their ages. The mean age of the children is 8 years old, and the standard deviation is 1.0526.
- [1] "Uniform Distribution" by Wikipedia
- [2] "Probability Density Function" by Wikipedia
- [3] "Mean and Standard Deviation" by Wikipedia
- [1] "Introduction to Probability and Statistics" by William Feller
- [2] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole
- [3] "Mathematical Statistics and Data Analysis" by John A. Rice
Suppose that the age of students at George Washington Elementary School is uniformly distributed between 5 and 11 years old
In our previous article, we explored the concept of a uniform distribution and how it applies to the age of students at George Washington Elementary School. We also discussed the process of selecting a random sample of 46 children from the school and analyzing their ages. In this article, we will answer some frequently asked questions related to uniform distribution and age analysis.
Q: What is a uniform distribution?
A: A uniform distribution is a type of probability distribution where every possible outcome has an equal chance of occurring. In the case of the age of students at George Washington Elementary School, we can assume that the age of each student is uniformly distributed between 5 and 11 years old.
Q: How do you calculate the probability density function (PDF) of a uniform distribution?
A: The probability density function (PDF) of a uniform distribution is given by the formula:
f(x) = 1 / (b - a)
where f(x) is the PDF, x is the age of the student, a is the lower bound of the distribution (5 years old), and b is the upper bound of the distribution (11 years old).
Q: What is the mean age of the 46 children?
A: The mean age of the 46 children is 8 years old, which is the expected value of the uniform distribution.
Q: What is the standard deviation of the ages of the 46 children?
A: The standard deviation of the ages of the 46 children is 1.0526, which is a measure of the spread of their ages.
Q: How do you calculate the variance of the ages of the 46 children?
A: To calculate the variance, you need to calculate the squared differences between each age and the mean, and then take the average of these squared differences.
Q: What is the significance of the uniform distribution in real-life scenarios?
A: The uniform distribution is significant in real-life scenarios where every possible outcome has an equal chance of occurring. For example, in a game of chance, the outcome of rolling a fair die is uniformly distributed between 1 and 6.
Q: How do you select a random sample from a population?
A: To select a random sample from a population, you need to use a random sampling method, such as simple random sampling or stratified random sampling.
Q: What are some common applications of uniform distribution?
A: Some common applications of uniform distribution include:
- Modeling the behavior of random variables
- Analyzing the distribution of data
- Making predictions about future events
- Evaluating the performance of a system
Q: What are some common mistakes to avoid when working with uniform distribution?
A: Some common mistakes to avoid when working with uniform distribution include:
- Assuming that the distribution is uniform when it is not
- Failing to account for outliers in the data
- Using the wrong formula for the PDF
- Not checking the assumptions of the uniform distribution
In this article, we answered some frequently asked questions related to uniform distribution and age analysis. We hope that this article has provided you with a better understanding of the concept of uniform distribution and how it applies to real-life scenarios.
- [1] "Uniform Distribution" by Wikipedia
- [2] "Probability Density Function" by Wikipedia
- [3] "Mean and Standard Deviation" by Wikipedia
- [1] "Introduction to Probability and Statistics" by William Feller
- [2] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole
- [3] "Mathematical Statistics and Data Analysis" by John A. Rice