Question

The following table shows economic development measured in per capita income PCINC.

a. What are the independent and dependent variables?

b. Draw a scatter plot.

c. Use regression to find the line of best fit and the correlation coefficient.

Short answer

Part a. Independent Variable: That does not get affected by another variable.

Dependent Variable:That gets affected by another variable.

Part b.

Part c. \(y=18.6615x - 34574.3225\)

\(r=0.9732\)

Step-by-step solution

Part a. Step 1. Explanation

Independent Variable:It refers to the variable that does not change/affected by any other variable.

Dependent Variable: It refers to the variable that gets change /affected by another variable.

E.g.If we take two variables like Age and height, then Age is an independent variable and height depends on age, so it is dependent variable.

Now, according to question,

“PCINC” is a dependent variable which depends on “YEAR”.

“YEAR” does not depend on any other variable, so we consider it as independent variables.

Part b. Step 1. Given information

Given table-

Part b. Step 2. Calculation

Scatter Plot:Scatter plot is graph which visualize or represents the two variables and we use dots to represent the points.

We plotted all the points by using the different dots. The graph is given below.

Part c. Step 1. Explanation

	YEAR(X)	PCINC(Y)			XY
	1870	340	3496900	115600	635800
	1880	499	3534400	249001	938120
	1890	592	3572100	350464	1118880
	1900	757	3610000	573049	1438300
	1910	927	3648100	859329	1770570
	1920	1050	3686400	1102500	2016000
	1930	1170	3724900	1368900	2258100
	1940	1364	3763600	1860496	2646160
	1950	1836	3802500	3370896	3580200
	1960	2132	3841600	4545424	4178720
SUM	19150	10667	36680500	14395659	20580850

A least square regression line is always is in the form of \(y=mx+c\), where m is slope of the line and \(c\) is \(y\) intercept.

\(m=\frac{n(\sum xy)-(\sum x)(\sum y)}{n(\sum x^{2})-(\sum x)^{2}}\)

\(m=\frac{10(20580850)-(19150)(10667)}{10(36680500)-(19150)^{2}}\)

\(m=18.6115\)

\(c=\frac{(\sum x)-m(\sum x)}{n}\)

\(c=\frac{10667-(18.6115)(19150)}{10}\)

\(c=-34574.3225\)

Now, putting the values of m and c in standard slope intercept form of line \(y=mx+c\).

\(y=18.6615x-34574.3225\)

Correlation Coefficient:

\(r=0.9732\)