Math Problem Statement
Solve this
Find the correlation coefficient and report it accurate to four decimal places.
Solution
To solve this, we need to compute the correlation coefficient for the given data set:
The data is:
x: & \quad 33, 25, 23, 30, 32, 39, 28 \\ y: & \quad 64, 70, 70, 66, 66, 61, 67 \end{aligned}$$ ### Formula for the Correlation Coefficient \[ r = \frac{n\sum(xy) - \sum x \sum y}{\sqrt{\left[n\sum x^2 - (\sum x)^2\right] \cdot \left[n\sum y^2 - (\sum y)^2\right]}} \] Where: - $$n$$ is the number of data points - $$\sum(xy)$$ is the sum of the product of $$x$$ and $$y$$ - $$\sum x$$, $$\sum y$$, $$\sum x^2$$, $$\sum y^2$$ are the respective sums. --- ### Step-by-Step Solution 1. **Calculate $$\sum x$$, $$\sum y$$, $$\sum x^2$$, $$\sum y^2$$, and $$\sum xy$$:** - For each $$x$$, calculate $$x^2$$ - For each $$y$$, calculate $$y^2$$ - For each pair $$(x, y)$$, calculate $$xy$$ I'll compute these values for you. numpy as np # Data points x = np.array([33, 25, 23, 30, 32, 39, 28]) y = np.array([64, 70, 70, 66, 66, 61, 67]) # Number of data points n = len(x) # Compute required sums sum_x = np.sum(x) sum_y = np.sum(y) sum_x2 = np.sum(x**2) sum_y2 = np.sum(y**2) sum_xy = np.sum(x * y) # Correlation coefficient formula components numerator = n * sum_xy - sum_x * sum_y denominator = np.sqrt((n * sum_x2 - sum_x**2) * (n * sum_y2 - sum_y**2)) # Correlation coefficient r = numerator / denominator rThe correlation coefficient $$r$$ is approximately $$-0.9826$$ (to four decimal places). This indicates a very strong negative linear relationship between the variables $$x$$ and $$y$$. --- Would you like details on any step, or have other questions? Here are some related questions you might explore: 1. How does the value of $$r$$ help interpret the strength of a relationship? 2. What happens if there is no linear correlation between $$x$$ and $$y$$? 3. How can we verify the calculations of $$r$$ step by step manually? 4. What is the coefficient of determination $$r^2$$, and how does it interpret the relationship? 5. How can this result be visualized using a scatterplot? **Tip:** Always plot your data to visually assess the relationship before relying solely on $$r$$.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Statistics
Correlation Coefficient
Linear Regression
Formulas
r = (n * Σ(xy) - Σx * Σy) / sqrt([(n * Σx² - (Σx)²) * (n * Σy² - (Σy)²)])
Theorems
Correlation Coefficient Formula
Suitable Grade Level
Grades 10-12
Related Recommendation
Calculate the Correlation Coefficient for Bivariate Data Set with Four Decimal Precision
Calculate the Correlation Coefficient for Given Data
Calculate Correlation Coefficient for x = [50, 60, 55, 65, 70] and y = [55, 65, 60, 70, 75]
Calculate the Correlation Coefficient for Given Data Points
Correlation Coefficient Calculation for Dataset (2, 2), (3, 4), (5, 4)