Question

Interest rates and mortgages. Since 1980 , average mortgage interest rates have fluctuated from a low of under $6\mathrm{\%}$ to a high of over $14\mathrm{\%}$. Is there a relationship between the amount of money people borrow and the interest rate that's offered? Here is a scatterplot of Total Mortgages in the United States (in millions of 2005 dollars) versus Interest Rate at various times over the past 26 years. The correlation is $-0.84$. a) Describe the relationship between Total Mortgages and Interest Rate. b) If we standardized both variables, what would the correlation coefficient between the standardized variables be? c) If we were to measure Total Mortgages in thousands of dollars instead of millions of dollars, how would the correlation coefficient change? d) Suppose in another year, interest rates were $11\mathrm{\%}$ and mortgages totaled $\mathrm{\$}250$ million. How would including that year with these data affect the correlation coefficient? e) Do these data provide proof that if mortgage rates are lowered, people will take out more mortgages? Explain.

Short answer

The relationship is strong and negative. Standardization or unit change does not affect correlation, which remains $-0.84$. Including the new point might slightly alter this. Correlation does not imply causation.

Step-by-step solution

Understand the correlation

The correlation coefficient given is $-0.84$. This indicates a strong negative linear relationship between Total Mortgages and Interest Rate. A negative correlation means as one variable increases, the other tends to decrease.

Standardization effect on correlation

Standardizing a variable involves rescaling it so that its mean is 0 and standard deviation is 1. However, standardization does not change the correlation coefficient value, so the correlation remains $-0.84$.

Effect of unit change on correlation

Changing units for Total Mortgages from millions to thousands does not affect the correlation coefficient because correlation is a unitless measure. Thus, the correlation remains $-0.84$.

Effect of additional data point

Including a new data point could change the correlation coefficient. The point $(11\mathrm{\%},\mathrm{\$}250\text{ million})$ seems consistent with an inverse relationship, potentially slightly weakening or leaving the correlation largely unchanged due to its proximity to data already affecting the correlation.

Correlation does not imply causation

The data showing a correlation of $-0.84$ does not prove causation. Therefore, while there appears to be a relationship, the data alone cannot prove that lowering mortgage rates will cause an increase in the amount of money people borrow.

Key concepts

Standardization Effect

Standardization is a statistical process used to make different data sets comparable by transforming them into a common scale. For each data point in our set, we subtract the mean and divide by the standard deviation. This process creates a standardized value with a mean of 0 and a standard deviation of 1.

Even though standardization might seem like a big change, it doesn't affect the correlation coefficient. The correlation measure is based on how the variables move together, not on their individual scales. Thus, when we standardize Total Mortgages and Interest Rate in our exercise, the correlation coefficient would still remain at $-0.84$.

• Standardized data has a mean of 0.
• Standardized data has a standard deviation of 1.
• Correlation remains unchanged post-standardization.

This is because correlation examines the strength and direction of a relationship, which isn't altered by simply changing the scale of measurement.

Unit Change Impact

The correlation coefficient is a value that expresses the degree to which two variables move in relation to each other. One key feature of this coefficient is that it is unitless.

This means that whether we measure Total Mortgages in millions or thousands of dollars, the correlation remains unchanged. Why? Because correlation examines the proportional relationships between variables. A change in units doesn't alter these proportional relations.

• Correlation is not dependent on units of measurement.
• Switching from millions to thousands will leave the correlation $-0.84$.
• This makes correlation a robust measure for comparing data sets with different units.

By focusing on how variables move together rather than their precise measurements, correlation provides a consistent insight into the relationship between variables, regardless of units.

Linear Relationship

A linear relationship is represented as a straight line graph where the change in one variable can be directly associated with a change in another. In our example, a strong negative correlation of $-0.84$ indicates a robust negative linear relationship between Total Mortgages and Interest Rate.

This negative correlation tells us that as interest rates increase, the amount of money borrowed tends to decrease. Conversely, as interest rates decrease, people tend to borrow more money.

• Visualized as a downward sloping line on a scatterplot.
• Strong linear relationships have correlation coefficients closer to $-1$ or $+1$.
• A negative coefficient like $-0.84$ means a decrease in one variable corresponds to an increase in the other.

Understanding linear relationships is crucial for predicting outcomes and exploring dependencies between variables.

Causation vs Correlation

While our analysis shows a strong negative correlation between mortgage rates and amounts borrowed, it's important to remember that correlation does not imply causation. This means that even though these variables are related, we cannot definitively say that a reduction in interest rates will directly result in more borrowing.

Other extraneous factors might contribute to this relationship, such as:

• Economic conditions
• Borrower confidence
• Availability of mortgage products

Therefore, while a negative correlation suggests a connection, further investigation and controlled experiments are needed to establish causation. Always consider the broader context and other possible variables that might be influencing the data at hand.