Below Is A Table Showing The Number Of Students Signed Up To Play Lacrosse After School In Each Age Group. Write An Equation Of The Regression Line For The Data And The Correlation Coefficient. Round To The Nearest Thousandth, If

Introduction

Regression analysis is a statistical method used to establish a relationship between two or more variables. In this case, we are interested in finding the relationship between the age of students and the number of students signed up to play lacrosse after school. The data is presented in a table, showing the number of students in each age group.

Table: Number of Students Signed Up to Play Lacrosse by Age Group

Equation of the Regression Line

To find the equation of the regression line, we need to calculate the slope (b) and the y-intercept (a) of the line. The slope represents the change in the dependent variable (number of students) for a one-unit change in the independent variable (age). The y-intercept represents the value of the dependent variable when the independent variable is zero.

Let's denote the number of students as Y and the age as X. We can calculate the slope and y-intercept using the following formulas:

b = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)² a = ȳ - b * x̄

where xi and yi are individual data points, x̄ and ȳ are the means of the independent and dependent variables, respectively.

Calculating the Slope and Y-Intercept

First, let's calculate the means of the independent and dependent variables:

x̄ = (10 + 12 + 14 + 16 + 18) / 5 = 14 ȳ = (15 + 25 + 35 + 45 + 55) / 5 = 32.6

Next, let's calculate the deviations from the means:

Now, let's calculate the slope and y-intercept:

b = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)² = (70.4 + 15.2 + 0 + 24.8 + 89.6) / (16 + 4 + 0 + 4 + 16) = 199.6 / 40 = 4.984

a = ȳ - b * x̄ = 32.6 - 4.984 * 14 = 32.6 - 69.856 = -37.256

Equation of the Regression Line

The equation of the regression line is:

Y = a + bX = -37.256 + 4.984X

Correlation Coefficient

The correlation coefficient (r) measures the strength and direction of the linear relationship between the independent and dependent variables. We can calculate the correlation coefficient using the following formula:

r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])

Let's calculate the correlation coefficient:

r = (70.4 + 15.2 + 0 + 24.8 + 89.6) / (√[16 + 4 + 0 + 4 + 16] * √[(-17.6)² + (-7.6)² + 2.4² + 12.4² + 22.4²]) = 199.6 / (√40 * √(306.56)) = 199.6 / (6.324 * 17.52) = 199.6 / 110.73 = 0.801

Conclusion

Q: What is regression analysis?

A: Regression analysis is a statistical method used to establish a relationship between two or more variables. In this case, we are interested in finding the relationship between the age of students and the number of students signed up to play lacrosse after school.

Q: What is the equation of the regression line?

A: The equation of the regression line is Y = a + bX, where Y is the number of students, X is the age, a is the y-intercept, and b is the slope. In this case, the equation is Y = -37.256 + 4.984X.

Q: What is the correlation coefficient?

A: The correlation coefficient (r) measures the strength and direction of the linear relationship between the independent and dependent variables. In this case, the correlation coefficient is 0.801, indicating a strong positive linear relationship between the age of students and the number of students signed up to play lacrosse.

Q: What does the slope (b) represent?

A: The slope (b) represents the change in the dependent variable (number of students) for a one-unit change in the independent variable (age). In this case, the slope is 4.984, indicating that for every one-year increase in age, the number of students signed up to play lacrosse increases by approximately 4.984.

Q: What does the y-intercept (a) represent?

A: The y-intercept (a) represents the value of the dependent variable (number of students) when the independent variable (age) is zero. In this case, the y-intercept is -37.256, indicating that when the age is zero, the number of students signed up to play lacrosse is approximately -37.256.

Q: What is the significance of the correlation coefficient?

A: The correlation coefficient indicates the strength and direction of the linear relationship between the independent and dependent variables. A correlation coefficient of 1 indicates a perfect positive linear relationship, while a correlation coefficient of -1 indicates a perfect negative linear relationship. A correlation coefficient close to 0 indicates no linear relationship.

Q: Can I use the regression line to predict the number of students signed up to play lacrosse for a given age?

A: Yes, you can use the regression line to predict the number of students signed up to play lacrosse for a given age. Simply plug in the age value into the equation Y = -37.256 + 4.984X to get the predicted number of students.

Q: What are some limitations of regression analysis?

A: Some limitations of regression analysis include:

• Assumption of linearity: Regression analysis assumes a linear relationship between the independent and dependent variables. If the relationship is non-linear, regression analysis may not be the best method.
• Assumption of independence: Regression analysis assumes that the observations are independent of each other. If the observations are not independent, regression analysis may not be the best method.
• Assumption of normality: Regression analysis assumes that the residuals are normally distributed. If the residuals are not normally distributed, regression analysis may not be the best method.

Q: What are some common applications of regression analysis?

A: Some common applications of regression analysis include:

• Predicting continuous outcomes, such as the number of students signed up to play lacrosse
• Analyzing the relationship between two or more variables
• Identifying the most important predictors of a continuous outcome
• Developing predictive models for continuous outcomes