Math Problem Statement
Annual low temperatures in a certain location have been tracked for several years. Let represent the year and the high temperature. Based on the data shown below, calculate the regression line (each value to four decimal places).
= + x
x y 4 14 5 12.81 6 11.02 7 13.33 8 8.14 9 9.25 10 9.16 11 6.27 12 4.58 13 4.29 14 2.4
Solution
To calculate the regression line (also known as the line of best fit) for the given data, we will use the formula for a linear regression line:
Where:
- is the dependent variable (temperature in this case).
- is the independent variable (year in this case).
- is the y-intercept of the line.
- is the slope of the line.
The slope and intercept are calculated as follows:
Where:
- is the number of data points.
- is the sum of the products of corresponding and values.
- , , , and are the respective sums of the -values, -values, squared -values, and squared -values.
I will calculate the values now.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, you can provide more data or ask a different question!
Would you like to attempt calculating the values manually? Here are some steps that can help you proceed:
- Calculate the sum of , , , and .
- Use the formulas provided earlier for (slope) and (intercept).
- Calculate the regression line with the values.
Let me know if you need more assistance or details! Here's how you might explore this further:
- What is the formula for the regression line?
- How do you calculate the slope and intercept ?
- What is the purpose of a regression line in data analysis?
- How can outliers affect the regression line?
- How would you interpret the slope and intercept in this context?
Tip: Double-check your calculations, especially when working with sums and squares.
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Statistics
Algebra
Formulas
y = b_0 + b_1 x
b_1 = [n(Σxy) - (Σx)(Σy)] / [n(Σx²) - (Σx)²]
b_0 = [Σy - b_1(Σx)] / n
Theorems
Least Squares Method
Suitable Grade Level
Grades 11-12
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