the population of a certain country from 1970 through 2010 is shown in the table to the right. use your…

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?\n\nx, number of years after 1969\n1 (1970)\n11 (1980)\n21 (1990)\n31 (2000)\n41 (2010)\n\ny, population (millions)\n203.8\n247.8\n252.4\n289.2\n316.8\n\nthe 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.)

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?\n\nx, number of years after 1969\n1 (1970)\n11 (1980)\n21 (1990)\n31 (2000)\n41 (2010)\n\ny, population (millions)\n203.8\n247.8\n252.4\n289.2\n316.8\n\nthe 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.)

Answer

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).