Microeconomics Study Guide: Scatterplots and Correlation

Q1. If the correlation between two variables is close to 0, what would a scatterplot show?

Background

Topic: Correlation and Scatterplots

This question tests your understanding of how the correlation coefficient ($r$) relates to the visual pattern in a scatterplot. Correlation measures the strength and direction of a linear relationship between two quantitative variables.

Key Terms and Formulas:

• Correlation coefficient ($r$): A value between -1 and 1 that indicates the strength and direction of a linear relationship.

• Scatterplot: A graph showing the relationship between two quantitative variables.

Step-by-Step Guidance

1. Recall that $r$ measures linear association. If $r$ is close to 0, it means there is little to no linear relationship between the variables.

2. Think about what a scatterplot would look like if there is no linear relationship: the points would not form a clear straight-line pattern.

3. Consider whether the points might show any pattern at all, or if they would just be scattered randomly.

4. Reflect on whether a non-linear pattern could still exist even if $r$ is close to 0.

Try solving on your own before revealing the answer!

Final Answer: b. a cloud of points with no visible pattern.

If the correlation is close to 0, the scatterplot will show a cloud of points with no visible straight-line pattern. This means there is no linear relationship between the variables, though other types of relationships may still exist.

Q2. Researchers found a correlation of $r = 0.3$ between soda consumed (in ounces) and aggression rating. If soda was measured in litres instead of ounces, what would the correlation be?

Background

Topic: Properties of Correlation

This question tests your understanding of how changing units of measurement affects the correlation coefficient. Correlation is a unitless measure, so it is not affected by linear transformations of the variables.

Key Terms and Formulas:

• Correlation coefficient ($r$): Measures linear association, unaffected by changes in units or linear scaling.

• Linear transformation: Changing units (e.g., ounces to litres) is a linear transformation.

Step-by-Step Guidance

1. Recall that correlation is calculated using standardized values (z-scores), which removes the effect of units.

2. Think about what happens when you convert ounces to litres: you multiply the values by a constant (1 litre = 35 ounces).

3. Consider whether multiplying all values of one variable by a constant changes the correlation coefficient.

4. Review the properties of correlation: it is unaffected by changes in scale or units.

Try solving on your own before revealing the answer!

Final Answer: b. 0.3

The correlation coefficient remains the same when units are changed by multiplying by a constant. Correlation is unitless and unaffected by linear transformations.

Q3. Find the correlation $r$ step by step for the sea surface temperature and coral growth data. Explain how your value for $r$ matches the direction of the linear pattern in your graph.

Background

Topic: Calculating Pearson Correlation

This question tests your ability to calculate the Pearson correlation coefficient ($r$) using data, and to interpret its sign and magnitude in relation to the scatterplot pattern.

Key Terms and Formulas:

• Pearson correlation coefficient ($r$):

$r = \frac{1}{n-1} \sum \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right)$

• $x_i, y_i$: individual data values

• $\bar{x}, \bar{y}$: means of $x$ and $y$

• $s_x, s_y$: standard deviations of $x$ and $y$

Step-by-Step Guidance

1. List the data for sea surface temperature ($x$) and coral growth ($y$).

2. Calculate the mean ($\bar{x}$, $\bar{y}$) and standard deviation ($s_x$, $s_y$) for each variable.

3. For each data point, compute $\frac{x_i - \bar{x}}{s_x}$ and $\frac{y_i - \bar{y}}{s_y}$.

4. Multiply these standardized values for each pair and sum the results.

5. Divide the sum by $n-1$ to get $r$.

6. Interpret the sign of $r$ in relation to the direction of the pattern in the scatterplot.

Try solving on your own before revealing the answer!

Final Answer: $r = -0.808$

The negative value of $r$ indicates a strong negative linear relationship, which matches the downward trend seen in the scatterplot.