You do an experiment on your pet fish, randomising which nights you turn on the aquarium light at night, and find that lights on at night results in the fish eating less. The relationship between the light being on and the amount the fish eats is:

Question

BlackTom AI · Accepted Answer

A negative correlation means that when one variable increases, the other tends to decrease; here, nights with the light on are associated with less food eaten.

1) Relevant knowledge points and concepts
- Correlation describes the direction and strength of a relationship between two variables (positive, negative, or none).  
- Randomisation in an experiment supports causal inference (reduces confounding), but the descriptive relationship observed is an inverse (negative) association if higher values of one variable align with lower values of the other.

2) Complete reasoning process step by step
- Represent the predictor (light) as a binary variable L (1 = light on, 0 = light off) and the response as amount eaten A.  
- If nights with L = 1 show lower A on average than nights with L = 0, the covariance between L and A is negative, implying a negative correlation.

3) Calculations with intermediate steps (illustrative example)
- Example data for 6 nights: L = [1,0,1,0,1,0], A = [2.5,4.0,2.0,5.0,1.5,4.5].  
- Means: mean(L) = 0.5, mean(A) = 3.25.  
- Deviations Li-mean(L): [0.5,-0.5,0.5,-0.5,0.5,-0.5]. Deviations Ai-mean(A): [-0.75,0.75,-1.25,1.75,-1.75,1.25].  
- Products: [-0.375,-0.375,-0.625,-0.875,-0.875,-0.625], sum = -3.75.  
- Sample covariance = -3.75/(n-1) = -3.75/5 = -0.75 (negative).  
- Sample standard deviations sd(L) ≈ 0.548, sd(A) ≈ 1.436, so correlation r = -0.75/(0.548*1.436) ≈ -0.95 (strong negative).

4) Why the correct answer is correct
- The negative covariance and negative correlation coefficient show that nights with the light on correspond to lower amounts eaten. That inverse relationship is by definition a negative correlation.

5) Why other multiple-choice possibilities are incorrect (brief)
- Positive correlation would mean lights on are associated with more eating (contrary to the data).  
- No correlation would mean no systematic change in eating with lights on (contrary to observed lower eating).  
- While randomisation also supports a causal conclusion (lights causing reduced eating), the descriptive term for the pattern observed is still a negative correlation.

6) Conclusion
The relationship between lights-on nights and amount eaten is a negative correlation, confirming the provided answer.

You do an experiment on your pet fish, randomising which nights you turn on the aquarium light at night, and find that lights on at night results in the fish eating less. The relationship between the light being on and the amount the fish eats is:单项选择题

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