Consider the following:

X

33

57

75

98

110

Y

201

224

181

129

111

1) What is slope of the regression line predicting Y from X,rounded to 2 decimal places? 

2) What is the intercept of the regression line predicting Y from X, rounded to 2 decimal places? 

3) What is the correlation between X and Y, rounded to 2 decimal places? 


To solve this, we will use simple linear regression, which fits a line to the data to predict \( Y \) based on \( X \). The equation of a regression line is:

\[
Y = a + bX
\]

Where:
- \( a \) is the **intercept**
- \( b \) is the **slope**
- The **correlation coefficient (r)** measures the strength and direction of the linear relationship.

The slope \( b \) is calculated as:

\[
b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2}
\]

The intercept \( a \) is calculated as:

\[
a = \frac{\sum Y - b(\sum X)}{n}
\]

Where \( n \) is the number of data points.

The correlation coefficient \( r \) is calculated as:

\[
r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n(\sum X^2) - (\sum X)^2][n(\sum Y^2) - (\sum Y)^2]}}
\]

I'll calculate the values now.1) The slope of the regression line predicting \( Y \) from \( X \) is \( -1.38 \).

2) The intercept of the regression line is \( 271.81 \).

3) The correlation between \( X \) and \( Y \) is \( -0.89 \), indicating a strong negative linear relationship.

Would you like any further details on the calculations or the interpretation?

Here are 5 follow-up questions to extend understanding:
1. What does the slope of \( -1.38 \) indicate about the relationship between \( X \) and \( Y \)?
2. How can we interpret the negative correlation of \( -0.89 \)?
3. What would happen to the line if the intercept were higher or lower?
4. Can we predict \( Y \) if \( X \) is outside the provided range?
5. What is the meaning of the residuals in a regression line?

**Tip:** A negative slope in regression implies that as the predictor variable increases, the response variable decreases.

Consider the following:

X
33
57
75
98
110

Y
201
224
181
129
111

1) What is the slope of the regression line predicting Y from X, rounded to 2 decimal places?
2) What is the intercept of the regression line predicting Y from X, rounded to 2 decimal places?
3) What is the correlation between X and Y, rounded to 2 decimal places?

Calculate the slope, intercept, and correlation between the given X and Y values: X = [33, 57, 75, 98, 110] and Y = [201, 224, 181, 129, 111]. This linear regression problem provides detailed steps and explanations for understanding the relationship between these variables, suitable for high school and introductory college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation