Question

(a) a yoga instructor collected student data. she found that students who had less sleep did not tend to have a greater range of motion during class. what can she 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)

To determine the conclusion, we analyze the relationship between sleep and range of motion. The data shows less sleep is not associated with greater range of motion, indicating a correlation (since a pattern exists, even if negative or no positive trend) and causation can't be assumed without further study. The first option is wrong (there is a correlation as a pattern is observed), the third assumes causation incorrectly. The second option correctly states there's a correlation (a relationship) and causation needs further study.

The study shows larger city population (variable A) relates to longer commute time (variable B), so there's a correlation. Causation (like population size directly causing longer commute) can't be confirmed without more analysis (e.g., other factors like infrastructure). The first option is wrong (correlation exists), the second incorrectly states no causation is present (we can't be sure, just that further study is needed). The third option correctly identifies correlation and that causation is uncertain without more study.

Employees starting work earlier don't necessarily drink more or less coffee, meaning there's no consistent relationship (correlation) between start time and coffee consumption. The second and third options assume a correlation exists, which is incorrect as the data shows no association. The first option correctly states no correlation.

Answer:

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.

Part (b)