A researcher collected two datasets. In Dataset A, the correlation between two measures was .60. In Dataset B, the correlation between two different measures was .30. Using this information, what can we conclude about the regression slopes in each dataset?

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Let's lay out what's being asked and the given information before evaluating each option.

- Question: A researcher has two datasets. In Dataset A, the correlation between two measures is .60. In Dataset B, the correlation between two different measures is .30. What can we conclude about the regression slopes in each dataset?
- Answer options:
  A. The slope is higher in dataset A than in dataset B
  B. The slope is significant in dataset A but not in dataset B.
  C. Both slopes are positive.
  D. None of the above.

Now, examine each option in turn:

Option A: The slope is higher in dataset A than in dataset B.
- Why this might seem plausible: A larger correlation (r = .60 vs r = .30) might be interpreted as implying a stronger relationship and thus a steeper slope.
- Why this is not necessarily true: The regression slope is not determined solely by the correlation coefficient. The slope of a simple linear regression of Y on X is given by b = r * (sy/sx), where sy is the standard deviation of Y and sx is the standard deviation of X. Since sx and sy can differ across datasets, a larger r does not automatically guarantee a larger slope. Without information about the standard deviations (or the units of the measures), we cannot conclude which slope is larger.

Option B: The slope is significant in dataset A but not in dataset B.
- Why this might be tempting: A larger correlation could lead to a more significant slope, depending on sample size and variability.
- Why this is not guaranteed: Significance depends on the standard error of the slope estimate, which in turn depends on sample size and the amount of noise in the data. A smaller correlation (r = .30) could still be statistically significant with a large sample size, and a larger correlation (r = .60) could be non-significant with a small sample size or high variability. Therefore significance cannot be inferred from the correlations alone.

Option C: Both slopes are positive.
- Why this is correct: In simple linear regression, the sign of the slope matches the sign of the correlation when predicting Y from X. Since both datasets have positive correlations (r = .60 and r = .30), the regression slopes must be positive in both datasets (assuming standard convention where the slope is calculated as the change in Y per unit change in X and the standard deviations are positive). The relationship is linear and positive in direction in both cases, so the slopes cannot be negative or zero based solely on the given correlations.
- Additional insight: The exact magnitudes of the slopes depend on the standard deviations of X and Y in each dataset, which are not specified. What we can assert from the correlation values is the direction of the relationship (positive), not the exact steepness.

Option D: None of the above.
- Why this is not correct: Since Option C accurately states that both slopes are positive, saying none of the above would be false. At least one correct statement exists, namely that both slopes are positive.

Summary:
- A cannot be concluded because slope magnitude depends on sy/sx in addition to r, and we lack that information.
- B cannot be concluded because significance depends on sample size and variability, not just the correlation value.
- C is correct because a positive correlation in both datasets implies positive regression slopes in both simple linear regressions.
- D is incorrect because there is a true statement among the options.

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A researcher collected two datasets. In Dataset A, the correlation between two measures was .60. In Dataset B, the correlation between two different measures was .30. Using this information, what can we conclude about the regression slopes in each dataset?

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