Topic 2.6: Regression and Interpolation Questions

Data is collected monthly for one year showing electricity use. A scatter plot shows a wave-like pattern repeating every 12 months. (a) Suggest the most appropriate model type. (b) Give one reason for your choice.

A student uses a quadratic model to predict the temperature at an Antarctic research station. The model was built using data for months 1 to 12. The student uses the model to predict the temperature in month 18. (a) Is this interpolation or extrapolation? (b) Give one reason why this prediction may not be reliable.

A researcher fits both a linear and an exponential regression to a dataset. The linear model gives R² = 0.74. The exponential model gives R² = 0.97. (a) State which model is more appropriate. (b) Give a reason for your choice.

A linear model C = 3.2t + 12 estimates the monthly cost (in hundreds of dollars) of running a factory, where t is the number of months since opening. Data was collected for 0 ≤ t ≤ 18. (a) Estimate the cost at t = 10. (b) Estimate the cost at t = 30. (c) State, with a reason, which estimate is more reliable.

An exponential model is fitted to population data collected from 1990 to 2020. The model is P(t) = 15000 · 1.03ᵗ, where t is years after 1990. (a) Estimate the population in 2010. (b) Estimate the population in 2040. (c) Give one reason why the estimate for 2040 may not be reliable.

A scatter plot of data shows a curve that rises steeply, passes near the origin, and never levels off. Two students suggest different models: one suggests exponential, the other suggests power. Give one reason for preferring the power model.

The model for sales revenue is R = −0.5x² + 30x − 200, valid for 10 ≤ x ≤ 50 (price in dollars). (a) Find the revenue when x = 25. (b) Find the price that maximises revenue. (c) A manager uses the model to predict revenue at x = 60. State one limitation of this prediction.

The table shows the number of hours studied x and the score y on a test for six students. Use the GDC to find (a) the linear regression equation, (b) the value of r, (c) the predicted score for a student who studied 7 hours.

Data for the speed v (m/s) and braking distance d (m) of a car are fitted using a power regression on a GDC. The GDC gives a = 0.045 and b = 2.03 for d = avᵇ. (a) Write the power model. (b) Find the braking distance when v = 20 m/s. (c) Comment on the reliability of your answer in (b) given the data covers v = 5 to 25 m/s.