Chapter 13 Reading Assignment
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Study guide for Chapter 13: Simple Linear Regression
13.1 Simple Linear Regression: Read this section and answer the following questions. ▪
What is an Independent Variable? What other name is it given? o
Ans: Independent Variables, otherwise known as explanatory variables, are variables that are added in a model to explain the variation in the dependent variable. The variable changed.
▪
Why is a Dependent Variable called by that name? o
Ans: Because it depends on the independent variable. It is the variable observed.
▪
What is the difference between Simple Regression and Multiple Regression? o
Ans: Simple Regression is the analysis of one independent variable and one dependent variable. Multiple Regression is the analysis of two or more independent variables on a dependent variable.
▪
What is the name of the equation used to explain the relationship between the Dependent and Independent Variables? o
Ans: The regression equation/model. A regression equation that gives a straight-line relationship between 2 variables in called a linear regression model while a regression model that gives anything else is a nonlinear regression model.
▪
What is a Simple Linear Regression Model? o
Ans: A model that gives a straight-line relationship between two variables. ▪
What is the form of equation a Simple Linear Regression model takes on? o
Ans: y= a + bx
▪
What is the coefficient of the x term in a linear equation called? o
Ans: The slope of the line. In y= 50 + 5x, 5 is the coefficient of x/slope. It is represented by the b value.
▪
What is the other piece of a linear equation, besides the x-term and its coefficient? o
Ans: The “a” term
. A represents the point at which the line intersects with the y-axis. In other words, the y-intercept is represented by a in y = a + bx. Y is the dependent variable.
▪
What is a Deterministic Model, and what does its equation look like? o
Ans: A deterministic model states that y is determined exactly by x, and for any given value of x, there is only one value of y. Represented by ▪
Does a Deterministic Model apply to every dependent/independent relationship? Why or why not? o
Ans: No, it only deals with 2 variables, the independent and dependent variables.
▪
If it doesn’t, what do we add to the right side of the regression equation?
o
Ans: We add the random error term, denoted by “ε”.
▪
What kind of model does this equation now represent? o
Ans: With this new term, we get the equation of a probabilistic regression model. ▪
Write the form of the entire equation of a Regression Model.
o
Ans: Y = A + Bx + ε
▪
What are the two reasons a random error term is added into the right side of a regression model equation? o
Ans: The random error term is added to encapsulate the effects of missing variables and to account for random variation.
▪
In the model equation on P. 550, what are A and B called? What is the model at the end of that page called? o
Ans: They are population parameters, representing the true values of the y-intercept and slope for a population regression line. The model for this equation is called the probabilistic model.
▪
What are the estimates of A and B found from sample data? What does the corresponding equation look like, and what is it called? o
Ans: Denoted by a and b, they are the values of the y-intercept and slope calculated from sample data on x and y. They make up the estimated regression model.
▪
What kind of graph should be used to plot the x and y values from a sample to try to visualize a relationship between those variables? o
Ans: A scatter diagram, otherwise known as a scatterplot.
▪
What is the name of the method used to find the equation of the line that will best fit the graph of the x and y values? What is the name of the line? o
Ans: The least squares method gives you the least squares regression line.
▪
What are the y-values from the actual data set called? o
Ans: The observed or actual value of y
▪
What are the y-values generated by the regression equation called? What symbol is used to represent them? o
Ans: The predicted value of y, it is represented by y hat.
▪
What is a residual? How is it represented in the Population Regression Model? o
Ans: ε
is also called the residual because it measures the surplus. It is the difference between the actual y value and the predicted y value.
▪
What symbol is used for its approximation in the Regression Model from the sample? What is this residual also called, and how is it found? o
Ans: e is used instead of ε
when using sample data. It is found by taking the actual y value –
y hat (the approximated y value).
▪
What is the sum of these errors always equal to? o
Ans: Equal to 0.
▪
What is Error Sum of Squares, how do we find it, and why do we find it? o
Ans: The Error Sum of Squares, denoted by SSE, is obtained by adding the squares of errors. We find it because we cannot minimize the sum or errors, so we minimize the error sum of squares.
▪
What are the regression equation and the values a and b now called, according to the first blue box on P. 551?
▪
Ans: The values of a and b that give the minimum SSE are called the least squares estimates of A and B, and the regression line obtained from these estimates is called the least squares regression line.
▪
What are all the equations used in finding the equation of this regression line (second blue box on P. 553)? Write each equation and any names given for them. o
Ans: This is only for sample regression line. a and b have same values but are calculated using these equations. SS is the sum of squares. End format will be y hat = a + bx. ▪
If we had population data available instead, how would we write the equation? o
Ans: If we have population data, we need to slightly tweak the equation. a is replaced by A, b and B, and n by N. The symbol μy|x
means the average values of y.
▪
Do the work for Example 13-1 and show it, so you can get experience doing a Simple Linear Regression. ▪
NOTE: YOU ARE USING THE SHORTCUT FORMULAS IN EXAMPLE 13-1, AND SHOULD ALWAYS USE THE SHORCUT FORMULAS WHEN DOING REGRESSION!!! THE BASIC FORMULAS ARE SHOWN IN A FOOTNOTE AT THE BOTTOM OF P. 508, BUT THEY ARE MORE DIFFICULT TO WORK WITH AND ARE NOT RECOMMENDED FOR REGULAR USE. ▪
Based on your reading of P. 554, interpret what the values of a and b from Example 13-1 represent in the context of the example. o
Ans: In the context of the example, the whole equation gives the regression of food expenditure on income. a, the y intercept, states that at 0 dollars income, you can estimate an average food expenditure of $150.15. b represents the amount of change for your different income levels, it is the slope.
▪
What is true about the linear relationship if b is positive? If it is negative? o
Ans: If b is positive, your line will travel up right. If it is negative, your line will travel down right.
▪
Read Case Study 13-1 and comment on why the value of A seems unrealistic. o
Ans: Because with the a value representing weight, if a player had a height of 0, their weight would be -690.5 pounds, which is impossible.
▪
Read Section 13.1.7 for your own educational benefit. ▪
From Section 13.1.8, what is Extrapolation and why can it be dangerous to do? o
Ans: Extrapolation is when you predict y values for a value of x that is out of its minimum and maximum limits. These values should not hold much value, as they are predictions. We assume that a linear relationship is still true between x and y, but we are not sure of the fact that they are. ▪
What is the other caution discussed in Section 13.1.8 regarding an attempt to do a Linear Regression on a data set?
We should make sure that our relationship is
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really linear, and this can be done by making a scatter diagram to view the trend of plots. If they are not in a straight line, it is not linear.
13.2 Coefficient of Determination: Read this section and answer the following questions. We will NOT deal with the materials in Subsection 13.2.1. ▪
What question could Coefficient of Determination help us answer about the regression equation? o
Ans: How well the independent variable describes the dependent variable in a regression model.
▪
What are Total Sum of Squares, what is the basic formula for calculating it, and what is the shortcut formula? What else is it equal to, from the formulas in Section 13.1? o
Ans: Equal to SSyy, used to find b in the least squares line, it is the total sum of squares, denoted by SST. Shortcut formula is:
Basic formula is:
▪
Redo all of the calculations in Table 13.4 for yourself, and show the work below. Reproduce the value of the Sum of Squared Errors at the bottom right of the table. ▪
What is the value of SST from Example 13-1? o
Ans: 125.71
▪
How is the Regression Sum of Squares calculated, and what does it represent?
o
Ans: SSR is the portion of SST described by the regression model. It is calculated by SSR = SST –
SSE.
▪
What does SSE now represent in this context? o
Ans: The error sum of squares. The SSE is the portion of SST that is not explained by the regression model.
▪
What are the symbol and formula for Coefficient of Determination (P. 567)? o
Ans: r^2 for sample data and p^2 for population data. Formula is SSR/SST
▪
What is the Computational Formula for Coefficient of Determination (top of P. 567)? o
Ans: ▪
Do the work for Example 13-3 and show it. ▪
Now, you are going to do a bit different calculation process. Calculate SSR using the formula SSR = (S
xy
)
2
/S
xx
. Report the result. 112.99. Work is included in the previous question response.
▪
Now, use the Regression Identity, SSE = SST –
SSR, on your results from Example 13-1 and compare this value of SSE to the value at the bottom of Table 13.4. (Note: the two estimates of Σe
2
and SSE should be equal, since they are the same thing.) Add the ideas from these last two bullet points to your notes.
Response in picture above. You will skip Section 13.3 and move to Section 13.4 now. 13.4 Linear Correlation: Read this section and answer the following questions. ▪
What does the Linear Correlation between two variables represent?
o
Ans:
The strength and direction of the linear associated between 2 variables. It is how closely the points in a scatter diagram are spread around the regression line and indicates if the relationship is positive or negative.
(NOTE: When we say two variables have a correlation, that does not mean the one is the dependent variable and the other is the independent. It only implies they seem to vary together in some way.) ▪
What is the symbol for Population Correlation Coefficient? For Sample Correlation Coefficient? o
Ans: Same as the coefficient of determination without the power of 2, so it is just r and p.
▪
Within what range of values does Linear Correlation Coefficient always fall? o
Ans: Always between -1 < p < 1 for population data. For sample data, it is: - < 1r < 1.
▪
What does it mean for two variables to have a Perfect Positive Linear Correlation? o
Ans: It means all the points on the scatter diagram are perfectly aligned with the line of the positive slope. ▪
What does it mean for two variables to have a Perfect Negative Linear Correlation? o
Ans: Same thing regarding the location of the points, but the line has a negative trend instead. ▪
What is the value of Linear Correlation Coefficient when two variables have absolutely NO correlation? o
Ans: r is close to 0, but never equal to 0.
▪
What is true about two variables when the value of Linear Correlation Coefficient between them is: o
a value like r = 0.95 Perfect positive linear correlation.
o
a value like r = -0.97 Perfect negative linear correlation
o
a value like r = 0.05 No correlation.
o
a value like r = -0.08 No correlation.
▪
Write the formula for Linear Correlation Coefficient. o
Ans: ▪
By what other name is Linear Correlation Coefficient known? o
Ans: The Pearson product moment correlation coefficient.
▪
Do Example 13-6 and show the work.
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13.5 Regression Analysis: A Complete Example: ▪
Do Example 13-8 parts (a)-(g) and show the work. You do not have to do the rest of the work in parts (h) and beyond.
You may stop reproducing work from Chapter 13 here. You may read other parts of this chapter and do the work for your added educational benefit if you choose to.