hw3soln - Biostatistics 100B Solutions To Homework...

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Biostatistics 100B Homework Solutions 3 January 22nd, 2007 Solutions To Homework Assignment 3 Warmup Problems (1) Basic Relationships: (a) There are many pairs of variables with positive linear relationships. For instance we saw in class that if C is temperature in degrees Celsius and F is temperature in degrees Fahrenheit then C and F have a perfect positive linear relationship and the equation of the model relating them is F = 32 + 1 . 8 C In general the equation for a linear relationship between two variables looks like Y = mX + b o rY = β 0 + β 1 X The first version is what you probably learned as the equation for a line in algebra. The second version is the way we will denote a straight line relationship when studying regression. If you have other examples of positive linear relationships you want to check with me, just send me an e-mail. (b) Let Y be the number of miles you have traveled in your car since your last visit to the gas station and let X be the number of gallons of gas you have left in your tank. Since you use gas as you travel, the farther you have gone the less gas you will have. This is a negative relationship and it is probably linear since your car uses gas at an approximately steady rate. If you have a car that gets about 30 miles per gallon and has a 10 gallon tank the equation would be roughly Y = 300 - 30 X I figured this out by noting that when you have no gas left you should have gone 10*30=300 miles and that you travel 30 miles for each gallon you use up. Of course the relationship won’t be perfect because your car does not get absolutely constant gas mileage! (c) A good example of a curved relationship is when Y is a company’s profits on a product and X is the price they charge for the product. Initially this is a positive relationship. If you sell the products at a price of 0 you will make no profit. (In fact you will lose money.) As you raise the price above 0 you will start bringing in more money. However, if you raise the price too much people will stop buying the product and your profits will go down again. If you plot profit versus price you will get an upside-down U-shape with a peak at the optimal sales price. (2) Regression Basics: (a) The simple linear model is Y = β 0 + β 1 X + ǫ . It is our model for all elements of the population and says that we expect Y to be linearly related to X with the typical value of Y for a given X being β 0 + β 1 X but allowing for individual variability, ǫ , about that typical value. The simple linear regression equation is E ( Y | X ) = μ Y | X = β 0 + β 1 X . It gives the average value of Y associated with a given X in the population. The estimated simple linear regression equation is ˆ Y = b 0 + b 1 X . It gives our best estimate of the simple linear regression equation based on sample data.
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