Step-by-Step Guidance for Stat1P99 Final Exam Practice Questions

Q1. Anscombe’s Data: Two data sets have almost the same r ≈ 0.816. Explain why reporting only r can be very misleading and what important feature you would miss without drawing scatterplots.

Background

Topic: Correlation and Data Visualization

This question tests your understanding of the limitations of the correlation coefficient (r) and the importance of visualizing data with scatterplots.

Key Terms and Concepts:

• Correlation coefficient (r): A measure of the strength and direction of a linear relationship between two variables.

• Scatterplot: A graphical representation of the relationship between two quantitative variables.

• Outlier: A data point that differs significantly from other observations.

Step-by-Step Guidance

1. Recall that the correlation coefficient $r$ measures the strength and direction of a linear relationship, but does not capture non-linear patterns or outliers.

2. Consider that two very different data sets can have the same $r$ value, even if their underlying relationships are not similar.

3. Think about what information a scatterplot provides that a single summary statistic like $r$ does not. For example, scatterplots can reveal non-linear relationships, clusters, or outliers.

4. Reflect on why relying solely on $r$ could lead to incorrect conclusions about the nature of the relationship between variables.

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Q2. Effect of Outlier on r: A scatterplot has 10 points with one clear outlier at (10,10). Describe how this outlier affects the value of r. What happens to r if the outlier is removed? What lesson does this teach?

Background

Topic: Sensitivity of Correlation to Outliers

This question examines how outliers can influence the correlation coefficient and the importance of checking for outliers in data analysis.

Key Terms and Concepts:

• Outlier: An observation that lies an abnormal distance from other values in a data set.

• Correlation coefficient (r): Sensitive to extreme values, especially in small data sets.

Step-by-Step Guidance

1. Recall that $r$ is calculated using all data points, so an outlier can have a large effect on its value.

2. Consider how the outlier at (10,10) might increase or decrease $r$, depending on its position relative to the trend of the other points.

3. Think about what would happen to $r$ if you removed the outlier: would $r$ become stronger (closer to 1 or -1) or weaker (closer to 0)?

4. Reflect on the broader lesson about the importance of examining data visually and not relying solely on summary statistics.

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Q3. Correlation vs Causation: Strong correlations are found between (i) storks and human births, (ii) pleasure boats and manatee deaths, (iii) cheese consumption and engineering PhDs. Does this mean one causes the other? Give a possible lurking variable for each.

Background

Topic: Correlation vs. Causation and Lurking Variables

This question tests your understanding of the difference between correlation and causation, and the concept of lurking variables.

Key Terms and Concepts:

• Correlation: A statistical association between two variables.

• Causation: When one variable directly affects another.

• Lurking variable: A variable not included in the analysis that influences both variables being studied.

Step-by-Step Guidance

1. Recall that correlation does not imply causation; two variables can be correlated due to a third, unmeasured variable.

2. For each example, think about what external factor (lurking variable) could be influencing both variables.

3. Consider how these lurking variables could create a spurious correlation between the two variables in each case.

4. Write down a plausible lurking variable for each pair.

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