A registered nurse is trying to develop a diet plan for patients. The required nutritional elements are the total daily requirements of each nutritional element are as indicated in table 2:

The nurse has four basic types to use when planning the menus. The units of nutritional elements per unit of food type are shown in the table below. Note than the cost associated with a unit of ingredient also appears at the bottom of table 3.

Moreover, due to dietary restrictions, the following aspects should also be considered when the developing the diet plan:
The chicken food type should contribute at most 25% of the total calories intake that will result from the diet plan.

The vegetable food type should provide at least 30% of the minimum daily requirements for vitamins.

Provide a linear programming formulation for the above case. (No need to solve the problem.)

...Linear -------------------------------------------------
Important
EXERCISE 27 SIMPLE LINEARREGRESSION
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 linkage between the theoretical statement and the equation is made prior to data collection and analysis. Linearregression is a statistical method of estimating the expected value of one variable, y, given the value of another variable, x. The term simple linearregression refers to the use of one independent variable, x, to predict one dependent variable, y.
The regression line is usually plotted on a graph, with the horizontal axis representing x (the independent or predictor variable) and the vertical axis representing the y (the dependent or predicted variable) (see Figure 27-1). The value represented by the letter a is referred to as the y intercept or the point where the regression line crosses or intercepts the y-axis. At this point on the regression line, x = 0. The value represented by the letter b is referred to as the slope, or the coefficient of x. The slope determines the...

...Introduction
This presentation on Regression Analysis will relate to a simple regression model. Initially, the regression model and the regression equation will be explored. As well, there will be a brief look into estimated regression equation. This case study that will be used involves a large Chinese Food restaurant chain.
Business Case
In this instance, the restaurant chain's management wants to determine the best locations in which to expand their restaurant business. So far the most successful locations have been near college campuses. This opinion is based on the positive numbers that quarterly sales (y) reflect and the size of the student population (x). Management's mindset is that over all, the restaurants that are within close proximity to college campuses with large student bodies generate more sales than restaurants located near campuses with small student bodies.
In the sample box below, xi is the size of the student population (in thousands) and yi is the quarterly sale (in thousands of dollars). The value for xi and yi for all of the 10 Chinese Food restaurants given in the sample are reflected as follows:
Sample Data:
(measured in 1,000s) (measured in $1,000s)
Restaurant Student Population Quarterly Sales
(i) (xi) (yi)
1 2 58
2 6 105
3 8 88
4 8 118
5 12 117
6 16 137
7 20 157
8 20 169
9 22 149
10 26 202
Methodology
Given the circumstances, "in...

...CHAPTER 1
INTRODUCTION
1.1 Background
Multiple linearregression efforts to model the relationship between two or more explanatory variables and a response variable by fitting a linear equation to observed data. Every value of the independent variable x is associated with a value of the dependent variable y. The population regression line for p explanatory variables x1, x2… xp is defined to be
μy=β°+β1x1+β2x2+ … + βpxp
This line describes how the mean response f changes with the explanatory variables. The observed values for y vary about their means μyand are assumed to have the same standard deviation σ. The fitted values b0, b1... bp estimate the parameters β1,β2,…,βp of the population regression line. Formally, the model for multiple linearregressions, given n observations, is
yi= β0 + β1xip + β2xip + ... βpxip + εi for i= 1,2, ... n.
In the least-squares model, the best-fitting line for the observed data is calculated by minimizing the sum of the squares of the vertical deviations from each data point to the line (if a point lies on the fitted line exactly, then its vertical deviation is 0). Because the deviations are first squared, then summed, there are no cancellations between positive and negative values. The least-squares estimates b0, b1, ... bp are usually computed by statistical software. The values fit by the equation b0 + b1xi1 + ......

...both sides of the linearregression line. When the incomes of the consumer increase the sales for cars also rises presenting a positive result. Therefore, as long as the incomes continue to grow the relationship to car sales will also trend to the right in an upward, positive motion.
B. What is the direction of causality in this relationship - i.e. does having a more expensive car make you earn more money, or does earning more money make you spend more on your car? In other words, define one of these variables as your dependent variable (Y) and one as your independent variable (X).
Depending on the each individuals perspective the variable can switch between dependent and independent based on the person’s viewpoint. For this purpose, the independent variable which is represented by (X) is the annual income. The dependent variable is represented by (Y) and is the cost of the car. The reason I chose to have the annual income as the independent variable is because a person will continue to look for a job with security, growth potential, and a higher income. The car is seen as a vehicle of transportation only and needed to get to work and home. It is a necessity, but not a luxury item with elaborate expenses. We can have the basic model without all the bells and whistles to accomplish the task to get to and from a location.
C. What method do you think would be best for testing the relationship between your dependent and independent variable,...

...How to Analyze the Regression Analysis Output from Excel
In a simple regression model, we are trying to determine if a variable Y is linearly dependent on variable X. That is, whenever X changes, Y also changes linearly. A linear relationship is a straight line relationship. In the form of an equation, this relationship can be expressed as
Y = α + βX + e
In this equation, Y is the dependent variable, and X is the independent variable. α is the intercept of the regression line, and β is the slope of the regression line. e is the random disturbance term.
The way to interpret the above equation is as follows:
Y = α + βX (ignoring the disturbance term “e”)
gives the average relationship between the values of Y and X.
For example, let Y be the cost of goods sold and X be the sales. If α = 2 and β = 0.75, and if the sales are 100, i.e., X = 100, the cost of goods sold would be, on average,
2 + 0.75(100) = 77. However, in any particular year when sales X = 100, the actual cost of goods sold can deviate randomly around 77. This deviation from the average is called the “disturbance” or the “error” and is represented by “e”.
Also, in the equation
Y = 2 + 0.75X + e
i.e.,
Cost of goods sold = 2 + 0.75 (sales) + e
the interpretation is that the cost of goods sold increase by 0.75 times the increase in sales. For example, if the sales increase by 20, the cost of goods sold...

...LINEARREGRESSION 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;...

...Tiffany Camp
ECO-250
Volker Grzimek
Regression Analysis of Work Hours in Relation to GPA
This research investigated the affects of working extra hours in a labor position on students’ GPAs each semester at Berea College. It was my belief that students who worked more hours were more likely to have lower GPAs due to their studying abilities and opportunities being compromised as a result of working too long (a negative correlation or trend between GPAs and hours worked each week). For each hour a student worked it was my belief that he or she became more fatigued, more stressed, and lost an hour in which to study. Each student must work at least ten hours here on campus as required by the Berea College Labor Program. Students may select to work more to make more money or to gain experience in a chosen field, or they may have to work more to meet work requirements for state assistance programs which help them financially but require that certain number of hours (usually 20) be worked by the student each week.
In order to test this hypothesis it was important that I collect unbiased samples. I did so by placing a survey in the labor program office where any random student was just as likely as any other to come in during this time of year when all students were turning in forms for their labor positions for the next year. I asked the students to record their classification (freshman, sophomore, etc.), whether or not they were recent...