Based On The Data Shown Below, Calculate The Regression Line (each Value To Two Decimal Places).$[ \begin{array}{|c|c|} \hline x & Y \ \hline 2 & 19.92 \ \hline 3 & 18.28 \ \hline 4 & 15.34 \ \hline 5 & 13.7 \ \hline 6 & 15.46 \ \hline 7 &
Introduction
Regression analysis is a statistical method used to establish a relationship between two or more variables. In this article, we will focus on calculating the regression line based on the given data. The regression line is a mathematical equation that describes the relationship between the independent variable (x) and the dependent variable (y). It is a powerful tool used in various fields, including economics, engineering, and social sciences.
Understanding the Data
The given data consists of two variables: x and y. The values of x range from 2 to 7, while the corresponding values of y range from 19.92 to 15.46. To calculate the regression line, we need to understand the relationship between these two variables.
Calculating the Mean of x and y
To calculate the regression line, we need to find the mean of x and y. The mean is calculated by summing up all the values and dividing by the number of values.
Mean of x = (2 + 3 + 4 + 5 + 6 + 7) / 6 = 4.5
Mean of y = (19.92 + 18.28 + 15.34 + 13.7 + 15.46 + 0) / 6 = 14.93
Calculating the Slope (b1)
The slope (b1) is a measure of the change in y for a one-unit change in x. It is calculated using the following formula:
b1 = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)²
where xi is the individual value of x, x̄ is the mean of x, yi is the individual value of y, and ȳ is the mean of y.
b1 = [(2-4.5)(19.92-14.93) + (3-4.5)(18.28-14.93) + (4-4.5)(15.34-14.93) + (5-4.5)(13.7-14.93) + (6-4.5)(15.46-14.93) + (7-4.5)(0-14.93)] / [(2-4.5)² + (3-4.5)² + (4-4.5)² + (5-4.5)² + (6-4.5)² + (7-4.5)²]
Simplifying the above equation, we get:
b1 = [(-2.5)(4.99) + (-1.5)(3.35) + (-0.5)(0.41) + (0.5)(-1.23) + (1.5)(0.53) + (2.5)(-14.93)] / [(-2.5)² + (-1.5)² + (-0.5)² + (0.5)² + (1.5)² + (2.5)²]
b1 = [-12.475 + (-5.025) + (-0.205) + (-0.615) + (0.795) + (-37.325)] / [6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25]
b1 = [-55.55] / [17.5]
b1 = -3.18
Calculating the Intercept (b0)
The intercept (b0) is the value of y when x is equal to zero. It is calculated using the following formula:
b0 = ȳ - b1x̄
b0 = 14.93 - (-3.18)(4.5)
b0 = 14.93 + 14.31
b0 = 29.24
Calculating the Regression Line
The regression line is a mathematical equation that describes the relationship between the independent variable (x) and the dependent variable (y). It is calculated using the following formula:
y = b0 + b1x
y = 29.24 + (-3.18)x
Conclusion
In this article, we calculated the regression line based on the given data. The regression line is a powerful tool used in various fields, including economics, engineering, and social sciences. It is a mathematical equation that describes the relationship between the independent variable (x) and the dependent variable (y). We calculated the slope (b1) and the intercept (b0) using the given data and then used these values to calculate the regression line. The regression line is a useful tool for predicting the value of y for a given value of x.
Final Answer
The final answer is:
y = 29.24 - 3.18x
Introduction
Regression analysis is a statistical method used to establish a relationship between two or more variables. In our previous article, we calculated the regression line based on the given data. In this article, we will answer some frequently asked questions related to regression line calculation.
Q1: What is the purpose of regression line calculation?
A1: The purpose of regression line calculation is to establish a relationship between the independent variable (x) and the dependent variable (y). It is a powerful tool used in various fields, including economics, engineering, and social sciences.
Q2: How is the regression line calculated?
A2: The regression line is calculated using the following formula:
y = b0 + b1x
where b0 is the intercept and b1 is the slope.
Q3: What is the slope (b1) in regression line calculation?
A3: The slope (b1) is a measure of the change in y for a one-unit change in x. It is calculated using the following formula:
b1 = Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)²
Q4: What is the intercept (b0) in regression line calculation?
A4: The intercept (b0) is the value of y when x is equal to zero. It is calculated using the following formula:
b0 = ȳ - b1x̄
Q5: What is the difference between simple linear regression and multiple linear regression?
A5: Simple linear regression is a type of regression analysis that involves two variables, x and y. Multiple linear regression is a type of regression analysis that involves more than two variables.
Q6: How do I interpret the results of regression line calculation?
A6: To interpret the results of regression line calculation, you need to understand the relationship between the independent variable (x) and the dependent variable (y). The slope (b1) indicates the change in y for a one-unit change in x, while the intercept (b0) indicates the value of y when x is equal to zero.
Q7: What are the assumptions of regression line calculation?
A7: The assumptions of regression line calculation are:
- Linearity: The relationship between x and y is linear.
- Independence: Each observation is independent of the others.
- Homoscedasticity: The variance of y is constant for all values of x.
- Normality: The distribution of y is normal.
Q8: How do I check the assumptions of regression line calculation?
A8: To check the assumptions of regression line calculation, you can use the following methods:
- Plot the residuals against the fitted values to check for linearity and homoscedasticity.
- Use the Shapiro-Wilk test to check for normality.
- Use the Durbin-Watson test to check for independence.
Conclusion
In this article, we answered some frequently asked questions related to regression line calculation. Regression analysis is a powerful tool used in various fields, including economics, engineering, and social sciences. It is a mathematical equation that describes the relationship between the independent variable (x) and the dependent variable (y). We hope that this article has provided you with a better understanding of regression line calculation and its applications.
Final Answer
The final answer is:
Regression line calculation is a powerful tool used in various fields, including economics, engineering, and social sciences. It is a mathematical equation that describes the relationship between the independent variable (x) and the dependent variable (y).