​(d) Comment on the type of relation that appears to exist between the weight of a car and its miles per gallon based on the scatter diagram and the linear correlation coefficient.
The variables weight of a car and its miles per gallon are 
▼ 
positively
not
negatively
 associated because r is 
▼ 
zero
negative
positive
 and the absolute value of the correlation coefficient is 
▼ 
greater
less
 than the critical value 
  
enter your response here.
​(Round to three decimal places as​ needed.)

To answer this question, we need to interpret the scatter diagram and the linear correlation coefficient (denoted as *r*), which measures the strength and direction of the relationship between two variables.

1. **Interpretation of the relationship**:
   - If the scatter diagram shows a **downward trend**, the relationship is **negative** (i.e., as weight increases, miles per gallon decreases).
   - If the scatter diagram shows an **upward trend**, the relationship is **positive** (i.e., as weight increases, miles per gallon increases).
   - If the scatter plot shows no clear trend, the relationship may be **zero** or very weak.

2. **Linear correlation coefficient (r)**:
   - If *r* is **positive** (i.e., close to +1), there is a positive correlation.
   - If *r* is **negative** (i.e., close to -1), there is a negative correlation.
   - If *r* is **close to 0**, there is little to no correlation.

3. **Critical value comparison**:
   - The absolute value of *r* should be compared to the critical value for a given significance level and sample size. If *|r|* is **greater** than the critical value, the correlation is statistically significant.
   - If *|r|* is **less** than the critical value, the correlation is not statistically significant.

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### Your question:

The variables **weight of a car** and **miles per gallon** are:
- **negatively** associated because *r* is **negative**, and the absolute value of the correlation coefficient is **greater** than the critical value (assuming statistical significance).

Would you like further clarification or help with any other steps (like calculating *r* or the critical value)?

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### Follow-up questions:
1. What does the critical value depend on in a correlation test?
2. How do you calculate the linear correlation coefficient (r)?
3. What does a **zero correlation** mean in terms of a scatter plot?
4. How does the sample size affect the critical value of *r*?
5. How can you determine whether a relationship is linear or non-linear based on a scatter plot?

**Tip**: Always ensure that you use the right significance level (usually 0.05 or 0.01) when comparing your *r* value to the critical value to determine whether the correlation is statistically significant.

Comment on the type of relation that appears to exist between the weight of a car and its miles per gallon based on the scatter diagram and the linear correlation coefficient.

Learn about the relationship between car weight and miles per gallon based on a scatter diagram and linear correlation coefficient. This problem explores the concept of correlation, with a focus on negative associations and statistical significance. Ideal for high school students in Grades 9-12.

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