Data are collected on the average second-hand price of a particular model of car, from back copies of car trading magazines. The model of car was manufactured in 2000.
Year 2001 2002 2003 2004 2005
Average price (£) 11015 5920 4123 3171 2587
Taking 2001 as year 1, 2002 as year 2, etc., enter the data into lists on your calculator and perform power regression on the data. Choose the three options which are true statements.
A The power regression model for the average price Y in year X of use is
Y = 11000 × X −0.899 ,with a and b correct to 3 significant figures.
B The power regression model for the average price Y in year X of use is
Y = 11000 × ( −0.899)X,with a and b correct to 3 significant figures.
C The regression model is a poor fit to the data.
D The regression model is a good fit to the data.
E The regression model predicts that in 2009 the average price of the car will be £1530 (to 3 significant figures).
F The regression model predicts that the car will be worth less than £1000 by 2014.
G The regression model predicts that the average price of the car will eventually become negative.
H The regression model is reliable up to 2020.
Carrie, you are asking for help for a specific function on your programmable calculator.
Unless somebody here has your type of calculator and is familiar with the particular function you want to use, I don't think you can expect much help with your problem.
To solve this problem, you need to perform power regression on the given data. Power regression is used to model data that follows a power law relationship, where one variable is a power of another variable.
To perform power regression on your calculator, follow these steps:
1. Enter the years (X-values) into one list, and the average prices (Y-values) into another list on your calculator.
2. Access the power regression function on your calculator. This function might be referred to as power regression, exponential regression, or curve fitting.
3. Enter the lists containing the X and Y values as the input for the power regression function.
4. The calculator will perform the regression analysis and provide you with the equation for the regression model.
Now let's evaluate the given statements using the power regression model:
A. The power regression model for the average price Y in year X of use is Y = 11000 × X^-0.899, with a and b correct to 3 significant figures.
To verify this statement, check if the equation provided matches the equation generated by the power regression on your calculator. If they match and the coefficients are correct to 3 significant figures, then statement A is true.
B. The power regression model for the average price Y in year X of use is Y = 11000 × (-0.899)^X, with a and b correct to 3 significant figures.
Similarly, check if this equation matches the one generated by the calculator. If they match and the coefficients are correct to 3 significant figures, then statement B is true.
C. To determine if the regression model is a good fit to the data, you can calculate the coefficient of determination (R^2). R^2 measures how well the regression model fits the data, with values ranging from 0 to 1. A value closer to 1 indicates a better fit. If the R^2 value is high, the regression model is a good fit, and if it is low, the regression model is a poor fit. Check the R^2 value from the regression analysis on your calculator to evaluate statement C.
D. Similarly, you can evaluate statement D by checking the R^2 value. If the R^2 value is high, the regression model is a good fit to the data.
E. To predict the average price in 2009 using the regression model, substitute X = 9 (2009-2000) into the equation generated by the regression analysis and check if the predicted value matches £1530 to 3 significant figures.
F. To predict the average price in 2014, substitute X = 14 (2014-2000) into the equation generated by the regression analysis and check if the predicted value is less than £1000.
G. Check the equation of the regression model and determine if it ever predicts negative average prices. If the equation does not produce negative values, then statement G is false.
H. The reliability of the regression model beyond 2020 cannot be determined without additional information. The regression model is based on the data provided and its predictive power is limited to the range of the data. It is not reliable beyond the available data unless there is a clear understanding of the underlying factors influencing the average price.
By following these steps and evaluating the statements using the power regression model and the available data, you will be able to determine which of the statements A to H are true.