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.