Question

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?
(a) there is no 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.
(b) 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.
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 may or may not be causation. further studies would have to be done to determine this.
a company found that employees who start work earlier do not necessarily drink more or less coffee. which statement is most likely true?
(c) 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:

(a) Just because students with less sleep have a greater range of motion doesn't mean one causes the other. Correlation doesn't imply causation. Further study could determine if there's a real relationship.
(b) A larger city population indicating longer commute time is a correlation, but no causation is established. More research would be needed to confirm a link.
(c) The fact that employees who start work earlier don't necessarily drink more or less coffee shows no clear - cut correlation between start time and coffee consumption. Further study could clarify.

Answer:

(a) 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.
(b) 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 may or may not be causation. Further studies would have to be done to determine this.
(c) There is no correlation between start time and amount of coffee consumed. There is a correlation between start time and amount of coffee consumed. However, there is no causation. This is because there is a correlation between start time and amount of coffee consumed with an earlier start time. However, there is no causation.