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:

y=b0+b1xy = b_0 + b_1 x

Where:

  • yy is the dependent variable (temperature in this case).
  • xx is the independent variable (year in this case).
  • b0b_0 is the y-intercept of the line.
  • b1b_1 is the slope of the line.

The slope b1b_1 and intercept b0b_0 are calculated as follows:

b1=n(xy)(x)(y)n(x2)(x)2b_1 = \frac{n(\sum{xy}) - (\sum{x})(\sum{y})}{n(\sum{x^2}) - (\sum{x})^2}

b0=yb1(x)nb_0 = \frac{\sum{y} - b_1(\sum{x})}{n}

Where:

  • nn is the number of data points.
  • xy\sum{xy} is the sum of the products of corresponding xx and yy values.
  • x\sum{x}, y\sum{y}, x2\sum{x^2}, and y2\sum{y^2} are the respective sums of the xx-values, yy-values, squared xx-values, and squared yy-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:

  1. Calculate the sum of xx, yy, xyxy, and x2x^2.
  2. Use the formulas provided earlier for b1b_1 (slope) and b0b_0 (intercept).
  3. 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:

  1. What is the formula for the regression line?
  2. How do you calculate the slope b1b_1 and intercept b0b_0?
  3. What is the purpose of a regression line in data analysis?
  4. How can outliers affect the regression line?
  5. 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