Math Problem Statement

What is the sample correlation coefficient for the given temperature and electricity use data? Use the provided data table and intermediate calculations to at least four decimal places, and round your final answer to three decimal places.

Solution

To solve the problem of determining the sample correlation coefficient (r) for the given data, let’s outline the steps:


Formula for Correlation Coefficient:

r=nxyxy(nx2(x)2)(ny2(y)2)r = \frac{n\sum xy - \sum x \sum y}{\sqrt{\left(n\sum x^2 - (\sum x)^2\right)\left(n\sum y^2 - (\sum y)^2\right)}}

Where:

  • nn: Number of data points
  • x\sum x: Sum of all xx-values (temperatures)
  • y\sum y: Sum of all yy-values (electricity usage)
  • xy\sum xy: Sum of the product of xx and yy for each row
  • x2\sum x^2: Sum of the squares of xx-values
  • y2\sum y^2: Sum of the squares of yy-values

Steps:

  1. Extract the given data:
    • Temperature (xx), Electricity Use (yy), and the product (xyxy).
  2. Calculate:
    • x,y,xy,x2,y2\sum x, \sum y, \sum xy, \sum x^2, \sum y^2.
  3. Plug the values into the formula and compute rr.
  4. Round the result to three decimal places.

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation
Linear Relationships

Formulas

Sample Correlation Coefficient Formula: r = (n * Σxy - Σx * Σy) / sqrt((n * Σx^2 - (Σx)^2) * (n * Σy^2 - (Σy)^2))

Theorems

Linear Correlation Coefficient Theorem

Suitable Grade Level

Grades 10-12