Question

question 10
the graph shows the population of puffins on a small island in the north pacific measured in years since 2000. the regression equation is ( p(t) = 4.5t + 14.13 ) with a coefficient of determination of ( r^2 = 0.872 ).

is it appropriate to use this model to estimate the puffin population in 2017?

a. yes, this is a strong correlation,
and the estimated puffin population
would be around 91 in 2017.
b. yes, this is a strong correlation and
( t = 17 ) is an interpolation value.
c. no, even though this is a strong
correlation, ( t = 17 ) is considered an
extreme extrapolation.
d. no, even though this is a strong
correlation, the trend would not
continue past ( t = 15 ).

graph with x - axis: 0, 2, 4, 6, 8, 10, 12, 14, 16; y - axis: 20, 40, 60, 80; red regression line and blue data points

Explanation:

1. First, analyze the coefficient of determination \( r^2 = 0.872 \), which is close to 1, indicating a strong linear correlation.
2. Then, determine the value of \( t \) for 2017: since \( t \) is years since 2000, \( t = 2017 - 2000 = 17 \).
3. Next, look at the graph's x - axis (t - values) which goes up to \( t = 16 \). Using \( t = 17 \) means we are predicting outside the range of the data used to create the model (extrapolation). Even with a strong correlation, extrapolating too far (especially when \( t = 17 \) is beyond the data's t - range shown) is not appropriate as the trend may not hold.
4. Now, evaluate the options:

• Option A: Claims it's appropriate, but extrapolating to \( t = 17 \) (beyond the data's t - range) is not appropriate just because correlation is strong. Also, the population estimate is a secondary issue to appropriateness.
• Option B: Says \( t = 17 \) is interpolation, but interpolation is within the data range. Since the data's t - range is up to 16, \( t = 17 \) is extrapolation, so B is wrong.
• Option C: Correctly states that even with strong correlation, \( t = 17 \) is extreme extrapolation, so using the model is not appropriate.
• Option D: There's no indication the trend stops at \( t = 15 \); the main issue is extrapolation to \( t = 17 \), not a supposed stop at \( t = 15 \).

Answer:

C. No, even though this is a strong correlation, t = 17 is considered an extreme extrapolation.