Math Problem Statement

Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is​ linear, determine whether it indicates a positive or negative association between the variables.

Use this information to answer the following.

020400102030ExplanatoryResponse

A scatter diagram has a horizontal axis labeled “Explanatory” from 0 to 40 plus in increments of 10 and a vertical axis labeled “Response” from 0 to 30 in increments of 5. The following 18 approximate points are plotted, listed here from left to right: (5, 29); (5, 26); (10, 25); (12, 24); (15, 24); (15, 21); (18, 22); (21, 21); (21, 18); (25, 19); (25, 16); (30, 16); (30, 13); (35, 14); (42, 10); (42, 8); (50, 6); (50, 4). The points generally fall from left to right at a constant rate.

Question content area bottom

Part 1

Do the two variables have a linear​ relationship?

A.

The data points do not have a linear relationship because they lie mainly in a straight line.

B.

The data points have a linear relationship because they do not lie mainly in a straight line.

C.

The data points have a linear relationship because they lie mainly in a straight line.

D.

The data points do not have a linear relationship because they do not lie mainly in a straight line.

Solution

The scatter plot described shows that the points generally fall from left to right at a fairly constant rate, suggesting that there is a linear relationship between the two variables.

Given that the points appear to follow a straight-line pattern, we can conclude that the relationship is linear. Additionally, since the points are decreasing from left to right (higher values of the explanatory variable correspond to lower values of the response variable), this suggests a negative association.

Thus, the correct answer is:

C. The data points have a linear relationship because they lie mainly in a straight line.

Would you like to explore more about linear associations or how to analyze scatter plots?

Here are five relative questions to deepen understanding:

  1. What is the difference between a positive and negative association in a scatter plot?
  2. How can we determine the strength of a linear relationship from a scatter plot?
  3. What role does the correlation coefficient play in assessing linear relationships?
  4. Can linear regression be applied to this data, and how would you calculate it?
  5. How do outliers affect the linearity and interpretation of a scatter plot?

Tip: In scatter plots, always check the overall trend and the dispersion of points to judge if a linear relationship exists and its direction (positive or negative).

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Math Problem Analysis

Mathematical Concepts

Linear Relationship
Scatter Plot
Negative Association

Formulas

Slope formula: (y2 - y1) / (x2 - x1)

Theorems

Linear Regression Theorem
Correlation Coefficient

Suitable Grade Level

Grades 9-11