Final answer:
The statement that two positively related variables cannot both decrease is false. In a positive correlation, both variables will increase and decrease together. The correct final answer is (b) false.
Step-by-step explanation:
The statement 'If two variables are directly (positively) related, they both can increase but they both cannot decrease' is false. When two variables are positively correlated, it indicates a direct relationship, meaning that if one variable increases, the other tends to increase as well, and conversely, if one variable decreases, the other tends to decrease. This relationship can be measured using a correlation coefficient, typically denoted by r. If r has a positive value, this signifies a positive correlation between the variables.
It is important to note that correlation does not necessarily imply causation. For example, there might be a positive correlation between the number of ice cream sales and crime rates during summer months, but this does not mean that ice cream sales cause crime to increase; both could be happening due to another factor, such as higher temperatures.
For directly proportional variables, represented by the equation y = kx, where k is the proportionality constant, a decrease in one variable results in a proportional decrease in the other. Thus, the initial claim in the question is incorrect; for variables with a positive correlation, both increases and decreases are possible.
So, the correct option in the final answer is (b) false.