Which one of the following correlations must be correct?

Question

BlackTom AI · Accepted Answer

Correlation coefficients measure the linear relationship between two quantitative variables and must lie between -1 and 1; the sign indicates direction (positive or negative) and the magnitude indicates strength.

1. Relevant knowledge points and concepts
- Pearson’s r applies to two numeric (quantitative) variables. Nominal categorical variables (like color, job type, major, or “favorite platform”) do not support a meaningful Pearson correlation unless they are properly and meaningfully numeric-coded.
- r ranges from -1 (perfect negative linear) to +1 (perfect positive linear); values near ±0.7 indicate a moderately strong linear relationship.
- The sign of r indicates whether one variable tends to increase as the other increases (positive) or decrease as the other increases (negative).

2. Step-by-step reasoning
- Examine each answer option to see whether both variables are quantitative:
  a. Outside temperature (quantitative) and color of the sky (categorical nominal) — not both numeric.
  b. Annual income (quantitative) and type of job (categorical) — type of job is nominal.
  c. Age (quantitative) and favorite social media type (categorical) — favorite type is nominal.
  d. Driving speed (quantitative) and amount of traffic (quantitative) — both numeric/quantifiable.
  e. Academic major (categorical) and number of credits (quantitative) — major is categorical.
- Only option (d) contains two quantitative variables, so only it can correctly be associated with a Pearson correlation coefficient.

3. Intermediate calculations / formula note
- Pearson r = [cov(X,Y)] / [sd(X) sd(Y)], which requires numeric X and Y. Since only driving speed and traffic amount provide numeric X and Y, r can be computed there. No numeric computation is necessary here because the question asks which correlation must be correct structurally.

4. Why the correct answer is correct
- Driving speed and amount of traffic are logically negatively related: as traffic increases, typical driving speed tends to decrease; r = –0.71 is a plausible moderately strong negative correlation and is the only choice that pairs two quantitative variables.

5. Why other options are incorrect
- Options (a), (b), (c), (e) pair a quantitative variable with a nominal/categorical variable, so reporting a Pearson correlation like 0.64, 0.82, –0.49, or 0.55 is not appropriate or meaningful without additional numeric encoding.

6. Conclusion
Therefore the correct choice is: “The correlation between driving speed and amount of traffic is r = –0.71.” This matches the provided answer.

Which one of the following correlations must be correct?单项选择题

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