Question

(a) a yoga instructor collected student data. he found that students who had less sleep did not tend to have a greater range of motion during class. what can he conclude?
there is no correlation between amount of sleep and range of motion.
there is a correlation between amount of sleep and range of motion. there may or may not be causation. further studies would have to be done to determine this.
there is a correlation between amount of sleep and range of motion. there is probably also causation. this is because there is likely an increase in the range of motion with an increase in the amount of sleep.
(b) a traffic study determined that a larger city population indicates a longer commute time for drivers. what can we say from this information?
there is no correlation between commute time and city population.
there is a correlation between commute time and city population. however, there is no causation. this is because there is an increase in the commute time with an increase in the city population.
there is a correlation between commute time and city population. there may or may not be causation. further studies would have to be done to determine this.
(c) a company found that employees who start work earlier do not necessarily drink more or less coffee. which statement is most likely true?
there is no correlation between start time and amount of coffee consumed.
there is a correlation between start time and amount of coffee consumed. there may or may not be causation. further studies would have to be done to determine this.
there is a correlation between start time and amount of coffee consumed. however, there is no causation. this is because there is probably an increase in the amount of coffee consumed with an earlier start time.

Explanation:

Part (a)

The instructor found a relationship (students with less sleep didn't tend to have greater range of motion, implying a negative correlation). Correlation doesn't imply causation, so further study is needed to check causation. The first option is wrong (there is a correlation), the third assumes causation too quickly. So the correct option is the second one.

A larger city population indicates longer commute time, so there's a positive correlation. But correlation doesn't mean causation (other factors could be at play), so we can't say there's no causation or that there is causation. The first option is wrong (there is a correlation), the second incorrectly says there's no causation (we can't be sure), so the third option is correct.

Employees starting work earlier don't necessarily drink more or less coffee, meaning no relationship (correlation) between start time and coffee consumption. The other options suggest a correlation, which is not supported by the data.

Answer:

B. There is a correlation between amount of sleep and range of motion. There may or may not be causation. Further studies would have to be done to determine this. (Note: Assuming the options are labeled as such; if original options had labels like A, B, C, adjust accordingly. Here, the second option in (a) is the correct one.)

Part (b)