Final answer:
The correlation coefficient measures the strength and direction of the linear relationship between two variables such as training and productivity. To have a coefficient of determination of at least 0.50, a correlation coefficient of approximately ±0.71 or more is needed. The significance of a correlation depends on the context and the size of the sample.
Step-by-step explanation:
The correlation between training and productivity refers to the strength and direction of a linear relationship between these two variables. It is measured by the correlation coefficient, typically represented by the symbol r. A strong positive correlation would suggest that as training increases, productivity also tends to increase, whereas a strong negative correlation would suggest that as training increases, productivity decreases.
For instance, a correlation coefficient of 0.5 (option b) indicates a moderate positive correlation. To answer the specific question about the necessary correlation to have a coefficient of determination of at least 0.50, we would need a correlation of approximately ±0.71 or more, since the coefficient of determination is the square of the correlation coefficient (r² = 0.50 implies r ≈ ±0.71).
The coefficient of determination helps to understand what percent of the variation in the dependent variable can be explained by the independent variable. When it comes to interpreting the significance of a correlation, the context and size of the sample must be considered. For a sample of 25 cases and a correlation of 0.45, one would conduct a t-test to determine if this correlation is statistically significant at a specified significance level, such as α = 0.05.
When organizing data to answer these types of questions, creating a table can oftentimes be an efficient way to visualize and compute necessary calculations, such as comparing the productivity of various countries.