Question

Emma is part of a team developing a new water softening agent. After gathering data about the softening agent's concentration in a volume of water over time, Emma recognizes this is an exponential decay situation and uses her computer to develop a model of the curve of best fit. The model of the curve of best fit for Emma's data has a correlation coefficient of r = -0.28. Which statement about Emma's model is true?

a) Predictions made using Emma's model are reliable because r is closer to 0 than -1.
b) Predictions made using Emma's model are unreliable because r is not positive.
c) Predictions made using Emma's model are unreliable because r is closer to 0 than -1.
d) Predictions made using Emma's model are reliable because r is not positive.

Answer 1

Emma's model for the water softening agent has a weak negative linear correlation with a correlation coefficient of -0.28. Therefore, predictions made using the model are unreliable because the r-value indicates a weak linear relationship.

Step-by-step explanation:

The correlation coefficient (r) is a statistical measure that shows the degree of linear correlation between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive linear correlation, -1 indicates a perfect negative linear correlation, and values closer to 0 suggest a weak or no linear relationship. If Emma's model of the water softening agent concentration has an r = -0.28, it indicates a weak negative linear correlation.

Answer c) 'Predictions made using Emma's model are unreliable because r is closer to 0 than -1' is the correct statement. This is because an r-value that is closer to 0, whether positive or negative, signifies a weaker linear relationship, thus making predictions from the model less reliable. A strong correlation, which is generally considered an r-value above 0.7 in absolute value, is absent here, making the model's predictive power questionable.