Linear Inequalities

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Bitsy Griffin

Technologie Business

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Linear Inequalities with both x & y ,[object Object],[object Object],[object Object]

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FSE 200 Adkins Page 1 of 10 Simple Linear Regression Correlation only measures the strength and direction of the linear relationship between two quantitative variables. If the relationship is linear, then we would like to try to model that relationship with the equation of a line. We will use a regression line to describe the relationship between an explanatory variable and a response variable. A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x. Ex. It has been suggested that there is a relationship between sleep deprivation of employees and the ability to complete simple tasks. To evaluate this hypothesis, 12 people were asked to solve simple tasks after having been without sleep for 15, 18, 21, and 24 hours. The sample data are shown below. Subject Hours without sleep, x Tasks completed, y 1 15 13 2 15 9 3 15 15 4 18 8 5 18 12 6 18 10 7 21 5 8 21 8 9 21 7 10 24 3 11 24 5 12 24 4 Draw a scatterplot and describe the relationship. Lay a straight-edge on top of the plot and move it around until you find what you think might be a “line of best fit.” Then try to predict the number of tasks completed for someone having been without sleep 16 hours. Was your line the same as that of the classmate sitting next to you? Probably not. We need a method that we can use to find the “best” regression line to use for prediction. The method we will use is called least-squares. No line will pass exactly through all the points in the scatterplot. When we use the line to predict a y for a given x value, if there is a data point with that same x value, we can compute the error (residual): Our goal is going to be to make the vertical distances from the line as small as possible. The most commonly used method for doing this is the least-squares method. The least-squares regression line of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible. Equation of the Least-Squares Regression Line · Least-Squares Regression Line: · Slope of the Regression Line: · Intercept of the Regression Line: Generally, regression is performed using statistical software. Clearly, given the appropriate information, the above formulas are simple to use. Once we have the regression line, how do we interpret it, and what can we do with it? The slope of a regression line is the rate of change, that amount of change in when x increases by 1. The intercept of the regression line is the value of when x = 0. It is statistically meaningful only when x can take on values that are close to zero. To make a prediction, just substitute an x-value into the equation and find . To plot the line on a scatterplot, just find a couple of points on the regression line, one near each end of the range of x in the data. Plot the points and connect them with a line. .

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