Question

the population of a certain country from 1970 through 2010 is shown in the table to the right. use your graphing utilitys linear regression option to obtain a model of the form y = ax + b that fits the data. how well does the correlation coefficient, r, indicate that the model fits the data?

x, number of years after 1969
1 (1970)
11 (1980)
21 (1990)
31 (2000)
41 (2010)

y, population (millions)
203.8
247.8
252.4
289.2
316.8

the model of the form y = ax + b that fits the data is y = 2.777x+201.029. (type integers or decimals rounded to three decimal places as needed.)

Explanation:

Step1: Recall correlation - coefficient concept

The correlation coefficient \(r\) measures the strength and direction of a linear relationship between two variables. A value of \(r\) close to \(1\) or \(- 1\) indicates a strong linear relationship, while a value close to \(0\) indicates a weak linear relationship. To find \(r\), we can use a graphing utility's linear - regression feature.

Step2: Input data into graphing utility

Input the data points \((x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)\) where \(x\) is the number of years after 1969 and \(y\) is the population (in millions) into the graphing utility. For example, the data points are \((1,203.8),(11,247.8),(21,252.4),(31,289.2),(41,316.8)\).

Step3: Use linear - regression function

Use the linear - regression function on the graphing utility. The utility will calculate the values of \(a\), \(b\) for the line \(y = ax + b\) (which we already have \(y=2.777x + 201.029\)) and also the correlation coefficient \(r\).

Answer:

The value of the correlation coefficient \(r\) needs to be calculated using a graphing utility with the given data points. Once calculated, if \(r\) is close to \(1\), it indicates a strong positive linear relationship (good fit of the model \(y = ax + b\) to the data), if \(r\) is close to \(0\), it indicates a weak linear relationship (poor fit of the model to the data). Without actually using the graphing utility, we cannot provide the exact value of \(r\).