The correlations between Age and Income as measured on 100 people is r = 0.72. Explain whether or not each of the possible conclusions below is justified.
a) When Age increases, Income increases as well.
b) The form of the relationship between Age and Income is straight.
c) There are no outliers in the scatterplot of Income vs. Age.
d) Whether we measure Age in years or months, the correlation will still be 0.72.
a) Choose the correct answer below.
A. No, because correlation does not imply causation.
B. Yes, because of the strength of the correlation and the size of the sample, it can be assumed that when Age increases, income increases as well.
C. No, because this is not known from the correlation alone. Extreme outliers can cause correlations to shift dramatically and can even cause a change in signs.
D. Yes, because the correlation is positive and close to 1, it can be assumed that when Age increases, income increases as well.
Answer (1)
Correlation of 0.72 between Age and Income means there's a tendency for these variables to increase together.
Correlation doesn't imply causation; other factors might influence both Age and Income.
The correlation coefficient measures the strength of a linear relationship but doesn't guarantee it's perfectly straight.
Correlation is unitless, so changing the units of Age (years to months) won't affect the correlation.
The correct answer is: No, because correlation does not imply causation.
Explanation
Analyze the problem and data We are given a correlation coefficient of r = 0.72 between Age and Income, measured on 100 people. We need to evaluate the validity of several conclusions based on this correlation.
Evaluate conclusion (a) Conclusion (a) states: When Age increases, Income increases as well. A positive correlation indicates a tendency for two variables to increase together. However, correlation does not imply causation. There might be other factors influencing both Age and Income. Therefore, we cannot definitively say that an increase in Age causes an increase in Income.
Evaluate conclusion (b) Conclusion (b) states: The form of the relationship between Age and Income is straight. The correlation coefficient r measures the strength and direction of a linear relationship. A correlation of 0.72 suggests a moderately strong linear relationship, but it doesn't guarantee that the relationship is perfectly straight. There could be some non-linearity present that the correlation coefficient doesn't fully capture.
Evaluate conclusion (c) Conclusion (c) states: There are no outliers in the scatterplot of Income vs. Age. The correlation coefficient can be significantly affected by outliers. A single outlier can either inflate or deflate the correlation. A correlation of 0.72 doesn't tell us whether outliers are present or not. We would need to examine the scatterplot to determine if any outliers exist.
Evaluate conclusion (d) Conclusion (d) states: Whether we measure Age in years or months, the correlation will still be 0.72. The correlation coefficient is a dimensionless quantity. It is not affected by changes in the units of measurement of the variables. Converting Age from years to months would simply multiply all Age values by 12, but this linear transformation would not change the correlation between Age and Income.
Final Answer for (a) Based on the analysis above, the correct answer for part (a) is: No, because correlation does not imply causation.
Examples
Understanding correlation is crucial in many fields. For example, in healthcare, a study might find a correlation between a certain lifestyle factor (like exercise) and a health outcome (like heart disease). While a positive correlation might suggest that more exercise is associated with a lower risk of heart disease, it doesn't prove that exercise directly prevents heart disease. Other factors, such as genetics or diet, could also play a significant role. This highlights the importance of considering other variables and not jumping to causal conclusions based solely on correlation.
