On warmer days, frogs tend to be more active and make more croaking noises. A biologist who studies frogs constructs a regression equation in order to predict number of frog croaks per minute based on temperature. If the regression equation explains 42.3% of the variability in number of frog croaks per minute, this means r is equal to approximately ______.

Note the answer you enter for this problem should be a number rounded to exactly two decimal places.

Question

On warmer days, frogs tend to be more active and make more croaking noises.  A biologist who studies frogs constructs a regression equation in order to predict number of frog croaks per minute based on temperature.  If the regression equation explains 42.3% of the variability in number of frog croaks per minute, this means r is equal to approximately ______.

Note the answer you enter for this problem should be a number rounded to exactly two decimal places.

BlackTom AI · Accepted Answer

R-squared (the coefficient of determination) is 42.3%, so r^2 = 0.423. The correlation coefficient r is the square root of r^2, with a sign that reflects the direction of the relationship.

1) Relevant concepts:
- r^2 measures the proportion of variance in the response explained by the predictor.
- r (Pearson correlation) satisfies r^2 = (r)^2 and lies between −1 and 1.
- r is positive when the association is positive (as temperature increases, croaks per minute increase), and negative when the association is negative.

2) Step-by-step reasoning:
- Convert percentage to decimal: 42.3% = 0.423.
- Take the square root to find the magnitude of r: |r| = sqrt(0.423).

3) Calculation with intermediate steps:
- Compute sqrt(0.423). Note that 0.65^2 = 0.4225, which is very close to 0.423.
- Therefore sqrt(0.423) ≈ 0.65038...
- Rounding to exactly two decimal places gives r ≈ 0.65.

4) Why this sign is positive:
- The problem statement says frogs are more active and make more croaking noises on warmer days, indicating a positive association between temperature and croaks per minute. Thus choose the positive square root rather than the negative one.

5) Why other potential answers are incorrect:
- Using 0.423 directly as r confuses r^2 with r; r must be between −1 and 1, and 0.423 as r is plausible but does not match the given r^2 (since 0.423^2 ≠ 0.423).
- Choosing −0.65 would give the correct r^2 but contradicts the described positive relationship (warmer → more croaks), so the sign would be wrong.

Conclusion: The correlation coefficient r is approximately 0.65, which matches the provided answer.

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