Question

MAKING AN ARGUMENT A set of data pairs has a correlation coeffi cient r = 0.3. Your friend says that because the correlation coeffi cient is positive, it is logical to use the line of best fi t to make predictions. Is your friend correct? Explain your reasoning.

Short answer

The friend's assertion is not entirely accurate. Although a positive correlation allows the use of a line of best fit for predictions, it does not ensure the accuracy of those predictions. In this case, the correlation coefficient is quite low (0.3), suggesting a weak positive relationship. Thus, any prediction based on this would have a fairly large margin of error.

Step-by-step solution

Understanding Correlation Coefficient

The correlation coefficient, symbolized as 'r', quantifies the degree to which two variables are related. A positive 'r' implies a positive relationship between the variables - as one increases, so does the other. However, the strength of the correlation depends on the absolute value of 'r'. If |r| is close to 1, the correlation is strong. If |r| is close to 0, the correlation is weak.

Interpreting r = 0.3

Given that 'r' is 0.3, this points to a weak positive relationship. While there is a direct relationship (as one variable increases, so does the other), the correlation is not strong. This means that while there is a trend, it is faint and there's a significant amount of scatter around the line of best fit.

Validity of the claim

While it is technically true that a line of best fit can be used for making predictions, one must consider how accurately those predictions reflect reality. Given the weak correlation (r=0.3), the line of best fit could have a substantial error in predictions, leading to inaccuracies. The positive correlation doesn't guarantee accurate predictions. It just indicates the direction of the relationship between the variables.

Key concepts

Positive Correlation

When we talk about "positive correlation," we're describing a situation where two variables tend to move in the same direction simultaneously. For example, as one variable increases, the other also tends to increase. This is indicated by a positive correlation coefficient, often symbolized as 'r'.

A positive value of 'r' tells us there is a positive relationship between the variables. However, the correlation coefficient also tells us about the strength of this relationship. It ranges from -1 to 1, where:

• '0' means no correlation — the variables don't relate.
• '1' is a perfect positive correlation — the variables rise together predictably.
• '-1' would be a perfect negative correlation where one variable increases as the other decreases.

So, a positive 'r' like 0.3 indicates that as one variable goes up, the other likely does too, though not strongly. This would mean there's a positive trend but with significant scatter, or variability, around that trend. In simple terms, the variables have a faint tendency to increase together.

Line of Best Fit

The "line of best fit" is a straight line that best represents the data on a scatter plot. This line is used to predict values, especially when examining relationships between variables. The line sat through the point clouds in a way trying to minimize the distance of all points from the line.

In the case of positive correlation, the line of best fit will have an upward slope, reflecting the positive association between the variables. However, just because this line can be drawn doesn't mean it provides highly reliable predictions.

• If 'r' is close to 1, the line of best fit is a good predictor.
• If 'r' is low, like 0.3, the line might not predict well as it serves only as a weak representation of the trend.

The line of best fit for an \( r = 0.3 \) typically has considerable scatter, or spread, meaning not all points are close to the line. This indicates a weak alignment of points and suggests a large potential margin of prediction error.

Predictive Accuracy

Predictive accuracy refers to how closely a model or line can predict actual outcomes based on given inputs. For making predictions with a line of best fit, the correlation coefficient 'r' significantly affects the accuracy.

When the value of 'r' is high (close to 1), the predictions tend to be more accurate as the points align closely with the line of best fit. However, a lower value like 0.3 only suggests a weak positive correlation and results in less accurate predictions.

• The weaker the correlation, the less precise the predictions.
• Consider the prediction's accuracy before making decisions based on weak 'r'.

Using a line of best fit with weak correlation could lead to predictions with large errors, which might be unreliable for practical purposes. Therefore, while the positive correlation allows using a line for prediction, the reliability of those predictions should be carefully evaluated.