Question

Income and housing. The Office of Federal Housing Enterprise Oversight (www.ofheo.gov) collects data on various aspects of housing costs around the United States. Here is a scatterplot of the Housing Cost Index versus the Median Family Income for each of the 50 states. The correlation is \(0.65\). a) Describe the relationship between the Housing Cost Index and the Median Family Income by state. b) If we standardized both variables, what would the correlation coefficient between the standardized variables be? c) If we had measured Median Family Income in thousands of dollars instead of dollars, how would the correlation change? d) Washington, DC, has a Housing Cost Index of 548 and a median income of about \(\$ 45,000\). If we were to include DC in the data set, how would that affect the correlation coefficient? e) Do these data provide proof that by raising the median income in a state, the Housing Cost Index will rise as a result? Explain.

Short answer

a) The relationship is positively correlated. b) The correlation remains 0.65. c) The correlation remains unchanged at 0.65. d) Including DC might decrease the correlation. e) No, correlation does not imply causation.

Step-by-step solution

Analyze the Relationship

The correlation between the Housing Cost Index and the Median Family Income is 0.65. This positive correlation indicates that, generally, as the median family income increases in a state, the housing cost index tends to be higher as well.

Correlation of Standardized Variables

The correlation coefficient remains the same when both variables are standardized. Therefore, the correlation between the standardized Housing Cost Index and the standardized Median Family Income is also 0.65.

Impact of Changing Units on Correlation

Changing the units of measurement for a variable (like measuring income in thousands of dollars instead of dollars) does not affect the correlation coefficient. Thus, the correlation would remain 0.65.

Including Washington, DC Data

Washington, DC, has a Housing Cost Index and median income that may be outliers compared to other states. Including DC could affect the correlation, potentially decreasing it if DC significantly deviates from the existing trend.

Causation versus Correlation

The data provided only show a correlation, not causation. Therefore, we cannot conclude that increasing the median income by itself will lead to a rise in the Housing Cost Index in a state, as other factors could be involved.

Key concepts

Scatterplot

A scatterplot is a great visual tool used in statistics to display the relationship between two variables in a dataset. In the context of the housing costs exercise, a scatterplot was used to plot the Housing Cost Index against Median Family Income for each state. Each point on the scatterplot represents a state, with the position of the point indicating the value of both variables for that state.

Here's how scatterplots can help you:

• Identify patterns: If the points form a clear trend, like a line, this indicates a relationship.
• Detect clusters: Groups of points can show subsets of the data that behave differently.
• Spot outliers: Points that don't fit the general pattern are easily identified in a scatterplot.

In our case, the scatterplot showed a positive correlation, meaning that as family income increases, so does the housing cost index—but keep in mind, it doesn't mean one causes the other to rise.

Standardized Variables

Standardized variables are used to compare datasets with different units or scales. To standardize, you subtract the mean from each data point and then divide by the standard deviation. This process converts values to a common scale with a mean of 0 and a standard deviation of 1.

In the housing exercise, standardizing the Housing Cost Index and Median Family Income doesn't change their correlation coefficient. Whether the data are in their original form or standardized, the correlation remains 0.65. This is because correlation measures the strength and direction of a linear relationship, which isn't affected by changes in scale or units.

Benefits of standardizing variables:

• Facilitates comparison: Makes diverse data more comparable.
• Prevents variable domination: Ensures no single variable has undue influence based on scale.
• Maintains correlation: Demonstrates that relationships are independent of metric units.

This technique is particularly useful when dealing with datasets involving different units of measurement.

Outliers

Outliers are data points that significantly differ from others in a dataset. They can influence the results of statistical analyses, including correlation. In the housing costs problem, Washington, DC was an outlier due to its exceptional Housing Cost Index and median income compared to other states.

Outliers can have a substantial impact on correlation:

• Skew results: An outlier can artificially inflate or deflate the correlation coefficient.
• Distort trends: Can misrepresent the true patterns within the data.
• Highlight uniqueness: May draw attention to special cases worth further investigation.

Including or excluding an outlier like Washington, DC could potentially lower the observed correlation if DC's data doesn't fit the established pattern. It's crucial to examine and understand outliers to accurately interpret data.

Causation vs Correlation

Causation implies that changes in one variable cause changes in another, whereas correlation simply indicates that two variables tend to move together. It's essential to differentiate between these to accurately interpret data.

In the housing costs exercise, the observed correlation of 0.65 suggests that states with higher median incomes tend to have higher housing costs. However, this correlation doesn't prove causation. Other factors, such as state policies, economic conditions, and housing availability, may influence both variables.

Key distinctions between causation and correlation:

• Correlation doesn't imply cause: Just because two variables correlate doesn't mean one causes the other.
• Consider confounding variables: Other variables might affect both of the variables in question.
• Use further analysis: Experiments or longitudinal studies may be needed to establish causation.

Being aware of these distinctions helps prevent misinterpretation of statistical relationships. It's crucial to conduct thorough analyses and consider additional evidence when inferring causation.