Math Problem Statement

Match these values of r with the accompanying​ scatterplots: negative 1​, negative 0.997​, negative 0.39​, 1​, and 0.749. Scatterplot 1 Scatterplot 2 Scatterplot 3 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 x y

A scatterplot has a horizontal x-axis labeled from 0 to 1 in increments of 0.1 and a vertical y-axis labeled from 0 to 8 in increments of 1. Ten points with varying vertical spread very loosely follow the pattern of a line that falls from left to right through the approximate points (0,5) and (1,4). 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 x y

A scatterplot has a horizontal x-axis labeled from 0 to 1 in increments of 0.1 and a vertical y-axis labeled from 0 to above 8 in increments of 1. Ten points almost strictly follow the pattern of a line that falls from left to right through the points (0.05, negative 8.23) and (0.88, negative 1.83). All coordinates are approximate. 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 x y

Match the values of r to the scatterplots. Scatterplot​ 1, requals ▼ negative 1 negative 0.997 0.749 negative 0.39 1 Scatterplot​ 2, requals ▼ 1 negative 0.997 negative 1 negative 0.39 0.749 Scatterplot​ 3, requals ▼ 1 negative 0.39 negative 0.997 0.749 negative 1 Scatterplot​ 4, requals ▼ negative 0.39 negative 0.997 negative 1 0.749 1 Scatterplot​ 5, requals ▼ negative 1 negative 0.997 1 0.749 negative 0.39 A scatterplot has a horizontal x-axis labeled from 0 to 1 in increments of 0.1 and a vertical y-axis labeled from negative 4 to 0 in increments of 1. Ten points strictly follow the pattern of a line that falls from left to right through the points (0.05, negative 0.20) and (0.95, negative 3.82). All coordinates are approximate. Scatterplot 4 Scatterplot 5 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 x y

A scatterplot has a horizontal x-axis labeled from 0 to 1 in increments of 0.1 and a vertical y-axis labeled from 0 to 8 in increments of 1. Ten points generally follow the pattern of a line that rises from left to right through the approximate points (0.1,1) and (0.9,7). The vertical spread of the points averages to approximately 2 units. 0 0.2 0.4 0.6 0.8 1 10 11 12 13 14 15 x y

Solution

To match the values of rr (correlation coefficient) with the scatterplots, we need to observe how tightly or loosely the points follow a line and whether that line has a positive or negative slope. Here is how to approach the problem:

  • r=1r = 1: Perfect positive linear correlation. The points lie exactly on a line with a positive slope.
  • r=−1r = -1: Perfect negative linear correlation. The points lie exactly on a line with a negative slope.
  • r=0.749r = 0.749: Strong positive linear correlation, but the points do not lie perfectly on a straight line.
  • r=−0.997r = -0.997: Strong negative linear correlation, close to r=−1r = -1, but the points may deviate slightly from a perfect line.
  • r=−0.39r = -0.39: Weak negative linear correlation. The points follow a general negative trend, but with more scatter.

Let's match each scatterplot with its rr-value:

Scatterplot 1:

  • The scatterplot shows points with a weak negative trend. The points loosely follow a line from left to right with a negative slope.
  • Best match: r=−0.39r = -0.39

Scatterplot 2:

  • The points closely follow a line with a steep negative slope. This suggests a strong negative linear correlation, but not perfect.
  • Best match: r=−0.997r = -0.997

Scatterplot 3:

  • The points follow a nearly perfect line with a negative slope.
  • Best match: r=−1r = -1

Scatterplot 4:

  • The points follow a nearly perfect line with a negative slope, but with slightly less precision than Scatterplot 3.
  • Best match: r=−0.997r = -0.997

Scatterplot 5:

  • The points follow a strong positive trend with some vertical spread.
  • Best match: r=0.749r = 0.749

Here is the final matching of the scatterplots:

  1. Scatterplot 1: r=−0.39r = -0.39
  2. Scatterplot 2: r=−0.997r = -0.997
  3. Scatterplot 3: r=−1r = -1
  4. Scatterplot 4: r=−0.997r = -0.997
  5. Scatterplot 5: r=0.749r = 0.749

Would you like more details or have any further questions?

Here are 5 related questions:

  1. How is the correlation coefficient rr calculated?
  2. What does an rr-value of 0 indicate about the relationship between variables?
  3. Can you have a strong nonlinear relationship with r≈0r \approx 0?
  4. What happens to rr if the scale of the data is changed?
  5. How does the size of the data set affect the reliability of rr?

Tip: Remember that the sign of rr indicates the direction of the relationship, while the magnitude shows the strength.

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation
Scatterplots

Formulas

Correlation Coefficient r = Σ((X - X̄)(Y - Ȳ)) / (nσXσY)

Theorems

Correlation Theorem
Linear Relationship Theorem

Suitable Grade Level

Grades 10-12

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