Math Problem Statement
South Dakota 8.1 25.1 Nevada 11.8 30.4 NewJersey 11.8 36.1 Texas 10.0 26.5 Kentucky 8.2 23.2 Maine8.4 25.0 Arizona 9.6 25.3 Wisconsin 10.2 27.4 Maryland 11.2 32.2 Michigan 10.4 27.8 Hawaii 11.5 27.8 Oregon 10.2 27.1 Minnesota 10.3 30.6 Utah 8.5 23.4
The least-squares regression line for these data has a slope of approximately 2.29 .
Answer the following. Carry your intermediate computations to at least four decimal places, and round your answers as specified below. (If necessary, consult a list of formulas.) (a)What is the value of the y-intercept of the least-squares regression line for these data? Round your answer to at least two decimal places.
(b)What is the value of the sample correlation coefficient for these data? Round your answer to at least three decimal places.
Solution
To solve these questions, we need to calculate the y-intercept of the least-squares regression line and the correlation coefficient using the given data. The slope has already been provided.
Let me break down the steps:
(a) Finding the y-intercept :
The equation of a regression line is: where is the slope and is the y-intercept. To calculate the y-intercept , we use the formula: Where:
- is the mean of the -values (first set of numbers).
- is the mean of the -values (second set of numbers).
- is the slope (which is given as 2.29).
We will now calculate the mean values of the and data, and use them to find .
Data:
- x-values: 8.1, 11.8, 11.8, 10.0, 8.2, 8.4, 9.6, 10.2, 11.2, 10.4, 11.5, 10.2, 10.3, 8.5
- y-values: 25.1, 30.4, 36.1, 26.5, 23.2, 25.0, 25.3, 27.4, 32.2, 27.8, 27.8, 27.1, 30.6, 23.4
Let me compute the mean and , and then find the y-intercept .It seems like I can’t do more advanced data analysis right now. Please try again later.
Feel free to try again or ask for clarifications on this problem! If you need help with another mathematical problem, let me know.
Here are some related questions you might be interested in:
- How can I calculate the slope of a regression line manually?
- What is the meaning of the correlation coefficient in regression analysis?
- How is the standard error of the estimate calculated for a least-squares regression line?
- What are the assumptions behind least-squares regression?
- How do outliers affect the least-squares regression line?
Tip: Always carry your calculations to a few more decimal places than you plan to round to, as this helps avoid rounding errors in intermediate steps.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Regression
Statistics
Correlation Coefficient
Formulas
y = mx + b (Equation of a regression line)
b = ȳ - m * x̄ (Formula for y-intercept)
Correlation coefficient formula
Theorems
Least Squares Regression
Pearson Correlation Coefficient
Suitable Grade Level
Grades 11-12, College
Related Recommendation
Calculate Least Squares Regression Line and Correlation Coefficient from Sample Data
Calculate the y-Intercept and Correlation Coefficient for Baseball Payroll Data
Sample Regression Equation and Correlation Coefficient Calculation
Calculating Correlation Coefficient, Slope, and Y-intercept from Data Points (6, 11), (8, 7), (7, 9)
Analyzing Linear Relationship Using Data Points x=2|4|6|8 and y=3|5|7|9