Answer :
When considering the correlation between two variables, it's important to understand how aggregation can affect the correlation coefficient. Here, we're comparing the correlation of two sets of SAT scores: mean scores at the state level and individual scores.
1. Understanding Correlation: Correlation measures the strength and direction of a linear relationship between two variables. A correlation of 1 indicates a perfect positive linear relationship, while a correlation of 0 indicates no linear relationship.
2. Aggregate Data vs. Individual Data: When data is aggregated, such as calculating mean scores at the state level, the variability within each state is reduced. This can lead to a higher correlation coefficient because the aggregated data tends to smooth out individual differences and emphasize the overall trend.
3. Interpreting the Given Scenario: We know that the correlation between mean 2018 Math SAT scores and mean 2018 ERW SAT scores for all 50 states and Washington D.C. is 0.985, which is already very high. Aggregated data, like state means, often produces higher correlations than individual-level data due to reduced variability and noise.
4. Expected Relationship: Given the high correlation at the mean state level, it is reasonable to expect that this correlation between state averages would generally be higher than the correlation seen at the individual level. This expectation is based on the principle that aggregation tends to reduce variability and highlight broader trends.
Therefore, one would expect the correlation between the mean state SAT scores for Math and ERW to be higher than the correlation between the scores of individuals on these tests.
1. Understanding Correlation: Correlation measures the strength and direction of a linear relationship between two variables. A correlation of 1 indicates a perfect positive linear relationship, while a correlation of 0 indicates no linear relationship.
2. Aggregate Data vs. Individual Data: When data is aggregated, such as calculating mean scores at the state level, the variability within each state is reduced. This can lead to a higher correlation coefficient because the aggregated data tends to smooth out individual differences and emphasize the overall trend.
3. Interpreting the Given Scenario: We know that the correlation between mean 2018 Math SAT scores and mean 2018 ERW SAT scores for all 50 states and Washington D.C. is 0.985, which is already very high. Aggregated data, like state means, often produces higher correlations than individual-level data due to reduced variability and noise.
4. Expected Relationship: Given the high correlation at the mean state level, it is reasonable to expect that this correlation between state averages would generally be higher than the correlation seen at the individual level. This expectation is based on the principle that aggregation tends to reduce variability and highlight broader trends.
Therefore, one would expect the correlation between the mean state SAT scores for Math and ERW to be higher than the correlation between the scores of individuals on these tests.
