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.

Explanation:

1. First, determine \( t \) for 2017: Since \( t \) is years since 2000, \( t = 2017 - 2000 = 17 \).
2. Analyze the graph's x - axis: The x - axis of the graph goes up to \( t = 16 \). So \( t = 17 \) is outside the range of the data used to create the regression model (the data points are within \( t \) values less than or equal to 16).
3. Recall the concept of extrapolation: Extrapolation is using a model to predict values outside the range of the original data. When \( t = 17 \), we are extrapolating. Also, the coefficient of determination \( r^{2}=0.872 \) (close to 1) indicates a strong linear correlation, but extrapolating too far (especially to an extreme value like \( t = 17 \) when the data only goes to \( t = 16 \)) is not appropriate.

• Option A: It says it's appropriate, but extrapolating to \( t = 17 \) (outside the data range) is not appropriate, so A is wrong.
• Option B: \( t = 17 \) is not an interpolation (interpolation is within the data range), so B is wrong.
• Option C: It correctly states that even with a strong correlation, \( t = 17 \) is an extreme extrapolation, so it's not appropriate to use the model for this estimate.
• Option D: There's no indication the trend stops at \( t = 15 \), the main issue is extrapolation to \( t = 17 \), so D is wrong.

Answer:

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