Question

which statement is true?
a strong association implies a positive correlation.
a positive correlation implies a strong association.
a positive correlation implies a positive slope.
a negative correlation implies a weak association.

Explanation:

• For option 1: A strong association can be either positive or negative. For example, a strong negative linear relationship between two variables (like the number of hours spent watching TV and the grade - point average in some cases) shows a strong association but a negative correlation. So, a strong association does not imply a positive correlation.
• For option 2: A positive correlation just means that as one variable increases, the other variable tends to increase. But the relationship can be very weak. For example, if we consider the number of books a person owns and their income, there might be a very weak positive correlation (not a strong association).
• For option 3: The formula for the sample correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\) and the formula for the slope of the regression line \(b = r\frac{s_{y}}{s_{x}}\) (where \(s_{x}\) and \(s_{y}\) are the sample standard deviations of \(x\) and \(y\) respectively). If \(r>0\), then \(b>0\) (since \(s_{x}>0\) and \(s_{y}>0\) for non - constant data sets). So, a positive correlation implies a positive slope.
• For option 4: A negative correlation means that as one variable increases, the other variable tends to decrease. But it can be a strong negative correlation. For example, the relationship between the speed of a car and the time taken to cover a fixed distance (assuming constant distance \(d = vt\), \(t=\frac{d}{v}\)) is a strong negative association (when \(d\) is fixed).

Answer:

A positive correlation implies a positive slope.