LINEAR REGRESSION MODELS W4315
HOMEWORK 2 ANSWERS February 15, 2010

Instructor: Frank Wood 1. (20 points) In the ﬁle ”problem1.txt”(accessible on professor’s website), there are 500 pairs of data, where the ﬁrst column is X and the second column is Y. The regression model is Y = β0 + β1 X + a. Draw 20 pairs of data randomly from this population of size 500. Use MATLAB to run a regression model speciﬁed as above and keep record of the estimations of both β0 and β1 . Do this 200 times. Thus you will have 200 estimates of β0 and β1 . For each parameter, plot a histogram of the estimations. b. The above 500 data are actually generated by the model Y = 3 + 1.5X + , where ∼ N (0, 22 ). What is the exact distribution of the estimates of β0 and β1 ? c. Superimpose the curve of the estimates’ density functions from part b. onto the two histograms respectively. Is the histogram a close approximation of the curve? Answer: First, read the data into Matlab. pr1=textread(’problem1.txt’); V1=pr1(1:250,1); V2=pr1(1:250,2); T1=pr1(251:500,1); T2=pr1(251:500,2); X=[V1;V2]; Y=[T1;T2]; Randomly draw 20 pairs of (X,Y) from the original data set, calculate the coeﬃcients b0 and b1 and repeat the process for 200 times b0=zeros(200,1); b1=zeros(200,1); i=0 for i=1:200 indx=randsample(500,20); x=X(indx); 1

y=Y(indx); avg x = mean(x); avg y = mean(y); sxx = sum((x − avg x).2 ); sxy = sum((x − avg x). ∗ (y − avg y)); b1(i) = sxy/sxx; b0(i) = avg y − b1(i) ∗ avg x; end; Draw histograms of the coeﬃcients b0 and b1 hist(b0) hist(b1)

Figure 1: Histogram of b0

Figure 2: Histogram of b1

2

i b. As we have known, b1 = i i(Xi −X)2 = i (Xii −X)2i = i Ki Yi whereKi = Xi −X¯ 2 ¯ ¯ i i i (Xi −X) So, b1 is a linear combination of Yi . Since Yi has a normal distribution, b1 also follows a normal distribution. E(b1 ) = i Ki E(Yi ) = i Ki (β0 + β1 Xi ) = i Ki β0 + ( i Ki Xi )β1 ¯ i (Xi −X) =0 ¯ i Ki = (Xi −X)2 i i i i i i i =1 ¯ 2 = ¯ 2 i Ki X i = i (Xi −X) i (Xi −X) E(b1 ) = 0 + 1 ∗...

...
Simple LinearRegressionModel
1. The following data represent the number of flash drives sold per day at a local computer shop and their prices.
| Price (x) | Units Sold (y) |
| $34 | 3 |
| 36 | 4 |
| 32 | 6 |
| 35 | 5 |
| 30 | 9 |
| 38 | 2 |
| 40 | 1 |
| a. Develop as scatter diagram for these data. b. What does the scatter diagram indicate about the relationship between the two variables? c. Develop the...

...Title - People Measurements in IB Math Studies
Introduction
The task was to gather the conclusive data from both 1st and 6th period IB Math Studies classes in the terms of each student’s separate data in the areas of height (measured in inches), shoe size, and arm span, also in inches.
Data
| Height in inches, x | Arm span in inches, y | x * y | x^2 |
| 64 | 66.5 | 4256 | 4096 |
| 63 | 68 | 4284 | 3969 |
| 63 | 64.7 | 4076.1 | 3969 |
| 61 | 63 |...

...Due in class Feb 6 UCI ID_____________________________
MultipleChoice Questions (Choose the best answer, and briefly explain your
reasoning.)
1. Assume we have a simple linearregressionmodel:
. Given a random sample from the population, which of
the following statement is true?
a. OLS estimators are biased when BMI do not vary much in the sample. ...

...LinearRegression deals with the numerical measures to express the relationship between two variables. Relationships between variables can either be strong or weak or even direct or inverse. A few examples may be the amount McDonald’s spends on advertising per month and the amount of total sales in a month. Additionally the amount of study time one puts toward this statistics in comparison to the grades they receive may be analyzed using the...

...Linear -------------------------------------------------
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STATISTICAL TECHNIQUE IN REVIEW
Linearregression provides a means to estimate or predict the value of a dependent variable based on the value of one or more independent variables. The regression equation is a mathematical expression of a causal proposition emerging from a theoretical framework. The...

...Linear-Regression Analysis
Introduction
Whitner Autoplex located in Raytown, Missouri, is one of the AutoUSA dealerships. Whitner Autoplex includes Pontiac, GMC, and Buick franchises as well as a BMW store. Using data found on the AutoUSA website, Team D will use LinearRegression Analysis to determine whether the purchase price of a vehicle purchased from Whitner Autoplex increases as the age of the consumer purchasing the vehicle...

...considers the relationship between two variables in two ways: (1) by using regression analysis and (2) by computing the correlation coefficient. By using the regressionmodel, we can evaluate the magnitude of change in one variable due to a certain change in another variable. For example, an economist can estimate the amount of change in food expenditure due to a certain change in the income of a household by using the regression...

...Algebra I Chapter 5 StudyGuide Writing Linear Equations
Name ________________
Due: Tuesday, January 17 (Exam week)
100 points
Writing Linear Equations in a Variety of Forms
Using given information about a __________, you can write an ________________of the line in _____________ different forms. Complete the chart:
Form (Name)
Equation
• •
Important information
The slope of the line is ____. The __ - ___________ of...

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