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 m=2.29m = 2.29 has already been provided.

Let me break down the steps:

(a) Finding the y-intercept bb:

The equation of a regression line is: y=mx+by = mx + b where mm is the slope and bb is the y-intercept. To calculate the y-intercept bb, we use the formula: b=yˉmxˉb = \bar{y} - m\bar{x} Where:

  • xˉ\bar{x} is the mean of the xx-values (first set of numbers).
  • yˉ\bar{y} is the mean of the yy-values (second set of numbers).
  • mm is the slope (which is given as 2.29).

We will now calculate the mean values of the xx and yy data, and use them to find bb.

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 xˉ\bar{x} and yˉ\bar{y}, and then find the y-intercept bb.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:

  1. How can I calculate the slope of a regression line manually?
  2. What is the meaning of the correlation coefficient in regression analysis?
  3. How is the standard error of the estimate calculated for a least-squares regression line?
  4. What are the assumptions behind least-squares regression?
  5. 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.

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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