Question

1. a set of data was used to create a linear, a quadratic, and an exponential regression model. the residual plots for the three models are shown above. based on the three residual plots, which of the following could be an appropriate model for the data?

(a) ( y = 3 + 2x )
(b) ( y = x^{2}+2x + 3 )
(c) ( y = 3(2)^{x} )
(d) ( y = 3 + 2log x )

1. mr. passwater loves to invest his money in mutual funds. over the past twenty years, he has closely tracked how his account grows and noticed that each year his account grows by approximately 10.4%. if mr. passwater wants to find a model the amount of money in his account over time, should he use a linear, quadratic, or exponential model for your answer.

Explanation:

Step1: Recall the property of residual plots

A good - fitting model has a residual plot with no apparent pattern. A linear model's residual plot should have points randomly scattered around the \(x\) - axis. A quadratic model \(y = ax^{2}+bx + c\) (where \(a
eq0\)) has a parabolic - shaped residual plot if it is a bad fit. An exponential model \(y = ab^{x}(a
eq0,b>0,b
eq1)\) has a residual plot that may show a trend if it is a bad fit.

Step2: Analyze the residual plots

The residual plot for the quadratic model shows a pattern. The residual plot for the linear model and the exponential model has points that are randomly scattered. But if we consider the growth rate information:
We know that the formula for compound - interest (which is similar to exponential growth) is \(A=P(1 + r)^{t}\), where \(r\) is the annual interest rate. Here, \(r=0.104\), so the amount of money \(y\) in the account after \(x\) years is of the form \(y = y_{0}(1 + 0.104)^{x}\). The general form of an exponential function is \(y = ab^{x}\), and in option (C) \(y = 3(2)^{x}\) (if we assume \(y_0 = 3\) and \(b = 2\) as a simple form of exponential growth model representation for the concept of growth).

Answer:

C. \(y = 3(2)^{x}\)