STATISTICS PROBLEM SOLVING

Statistics Problem Solving



Statistics Problem Solving

Introduction

Regression analysis uses a mathematical equation to describe this kind of relationship. In linear regression the equation describes the straight line which, on the average, provides the closest "fit" to all the points at once. There is only one such line and it is known as the least-squares regression line. It gets its name from the fact that if we measure the vertical distance (called a residual) between the line and each point on the graph, square each distance, and adds them up; the total will be smaller for this line than for any other. Hence, it is the line that best "fits" the data.

The general form of the equation is Y = a + bX, where Y is the predicted value of the dependent variable and X is the value of the independent variable. The caret (^) over the Y indicates that it is a predicted rather than an actual value. The regression constant (also known as the Y-intercept), which is represented by a in the equation, tells us the value of Y when X has a value of zero. It is also the point where the line crosses the vertical (Y) axis. The regression coefficient (or slope), which is represented by b in the equation, tells us how much the value of Y changes for each change in X.

A more complicated form of regression analysis is multiple regressions, where two or more independent variables are used simultaneously to estimate the value of a dependent variable (Fox, 2001). To estimate personal income, for example, we might use independent variables such as years of schooling (X1), occupational prestige score (X2), and gender (X3). The equation would then look like this:

Y=a+b1X1+b2X2+b3X3

For a relationship that looks more like a curve than a straight line, curvilinear regression may be used. In general, curvilinear techniques are more complicated to use and interpret and are less widely used in sociology than linear techniques (Lewis-Beck, 2000).

1. Find the equation of the regression line for the given data. 

Predict the value of Y when X = -2?

Predict the value of Y when X = 4?

X

-5

-3

4

1

-1

-2

0

2

3

-4

Y

-10

-8

9

1

-2

-6

-1

3

6

-8

Solution

The regression equation for the above given data set can be written as follows:

Y = -0.5515 + 2.0969 * (X)

 

Coefficients

Standard Error

t Stat

P-value

Intercept

-0.551515152

0.313354433

-1.760036217

0.116440376

X

2.096969697

0.107479684

19.51038206

4.94958E-08

It can be suggested that a unit increase in the value of X can increase the magnitude of Y by 2.0969 units, the magnitude of the slope i.e. 2.0969 is significant as its p-value i.e. 4.949E-08 is less than 0.05 which is the level of significance. The intercept has a negative effect on the overall regression equation, however this magnitude is insignificant as its p-value is greater than the level of significance.

Predicted Values

For X= -2

Y = -0.5515 + 2.09698 * (-2)

Y =

-4.745454545

For X = 4

Y = -0.5515 + 2.09698 * (4)

Y =

7.836363636

2. The data below are the final exam scores of 10 randomly selected statistics students and the number of hours they studied for the exam. Find the equation of the regression line for ...