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1.3 Approximate Linear Models Question 1: Which linear model is best? Question 2: How do you use linear regression functions? Question 3: How good is the linear model? In section 1.2, we found the demand function D(Q) for a dairy. The scatter plot and corresponding linear function are pictured in Figure 1. Figure 1 – The demand function D(Q) = -0.05Q + 7.75 for the data in Example 7 of section 1.2. In this scatter plot, the function passes through each point perfectly. In other words, the prices at each quantity in the data match the prices from the function at the corresponding prices. This occurred because the slopes between adjacent ordered pairs were exactly the same. In this section we’ll find linear models for data that do not lie along a straight line. 1 Question 1: Which linear model is best? In real life, data are not so perfect. A more common situation might be the data in Table 1. Table 1 Weekly Demand for Milk (thousands of gallons) 85 95 105 115 125 Average Price Per Gallon (dollars) 3.89 2.78 3.01 3.23 1.69 If we were to calculate the slope between adjacent points, we would find that the slope is not constant. This is easy to see if we graph the ordered pairs from the table in a scatter plot. Figure 2 – A scatter plot of the data in Table 1. The slope between adjacent ordered pairs is not constant. Although we cannot find a linear function that passes though all of the points, we can find a linear function that passes close to the points. This function is said to model the 2 data. A mathematical model is a representation of a real world system. In this case, the model is a representation of the relationship between the average price of a gallon of milk and the quantity of milk sold per week. Figure 3 – Two different linear functions that model the relationship between the average price of a gallon of milk and the number of gallons sold at that price. The linear functions in Figure 3 both pass close to the points, but which function passes “closest” to the points? Closeness is measured vertically from the data point to the line. For instance, let’s draw a vertical dashed line from each point to the linear function in Figure 3a. 3 Figure 4 – The dashed red lines represent the vertical distances between the data values and the linear function. The vertical distance between any data value and the line can be computed by subtracting the corresponding prices. To find the largest vertical distance at Q 115 , we need to find the average price from the data and from the linear model: Price at Q 115 from data: P 3.23 Price at Q 115 from linear function: Pˆ 0.05 115 8 2.25 The symbol P̂ indicates that the price has been estimated from the model whereas P indicates a data price. The difference between the data price and the linear model price is called the error or residual. For this price, the error is E P Pˆ 3.23 2.25 0.98 Since the error is positive, we know that the data price is higher than the model’s price. If the error is negative, the model’s price is higher than the data price. This is the case at Q 95 : 4 Price at Q 95 from data: P 2.78 Price at Q 95 from linear function: Pˆ 0.05 95 8 3.25 E P Pˆ 2.78 3.25 0.47 We can carry this process at each of the quantities in Table 1 and label it on a scatter plot. Figure 5 – A scatter plot of the data and the linear model. The dashed red line and red numbers indicate the error between the data and the model. If we want a linear function to go as close to the ordered pairs as possible, we need to collectively make these errors as small as possible. This is done by summing the errors at each quantity. If the sum of the errors is zero, the linear function might go through each point. 5 Figure 6 – In a, each data point coincides with the linear model so the sum of the errors is zero. In b, the sum of the errors is also zero, but the data points do not lie on the graph of the linear model. But we can’t simply sum the errors as is since some of the errors are positive and some are negative. By adding the errors together we might have some cancellation and be deceived into thinking that a linear model coincides with the data when it does not. This situation is illustrated in Figure 6. In graph a of Figure 6, each ordered pair of the data lies on the linear model so there is no error between the model and the data. The sum of the errors is zero. In this case, the linear model fits the data perfectly. But simply adding the errors is inadequate. In graph b of Figure 6, the sum of the errors is also zero. However the model does not fit the data perfectly. Some ordered pairs are above the linear model and others are below the linear model. For each positive error, there is a negative error that cancels it. Even though the sum is zero, the linear model is not a very good fit for the data. A better criterion for determining the fit is needed. Instead of simply summing the errors, sum the square of the errors. This eliminates the potential cancellation of the positive and negative errors. 6 Table 2 Q P P̂ P Pˆ P Pˆ 85 3.89 3.7500 0.1400 0.0196 95 2.78 3.2500 -0.4700 0.2209 105 3.01 2.7500 0.2600 0.0676 115 3.23 2.2500 0.9800 0.9604 125 1.69 1.7500 -0.0600 0.0036 2 Sum = 1.2721 In Table 2, the first three columns describe the quantity Q, the price P and the estimate of the price P̂ based on the linear model. The fourth column describes the error in the estimate and the last column corresponds to the squared error. For this model, the sum of the squared errors is 1.2721. A model that has a lower sum of squared error would be considered a better model. Let’s look at the model in Figure 3b, P 0.0395Q 7.0675 . Table 3 Q P P̂ P Pˆ P Pˆ 85 3.89 3.7100 0.1800 0.0324 95 2.78 3.3150 -0.5350 0.2862 105 3.01 2.9200 0.0900 0.0081 115 3.23 2.5250 0.7050 0.4970 125 1.69 2.1300 -0.4400 0.1936 2 Sum = 1.0174 7 This model, rounded to four decimal places, is better since the sum of the squared errors is lower. The model P 0.0395Q 7.0675 in Table 3 is the very best model for the data. This means that it has the lowest sum of squared errors. If we were to vary the slope and intercept in the model, no other combination would lead to a sum lower than 1.0174. The process of calculating the linear model with the smallest sum of squared errors is called linear regression. The model obtained through linear regression goes by several different names. Best linear model, least squares linear model and best line are all terms that can be used when referring to the model. The model is typically obtained using technology like a graphing calculator or Excel. The formulas required for calculating the slope and intercept of the best linear model without technology uses calculus and are beyond the scope of this text. 8 Question 2: How do you use linear regression functions? Once you have found a linear model using linear regression, you can use it to analyze problems. These problems may involve questions about the slope or the variables. Example 1 Use a Linear Model The least squares linear model for the data in Table 1 is P 0.0395Q 7.0675 . a. Use this model to determine the average price when 100 thousand gallons of milk are sold. Solution This equation relates Q, the quantity of milk sold in thousands of gallons, to the average price P. Since we are given a value of Q 100, substitute this into the model: P 0.0395 100 7.0675 3.12 At a price of $3.12, 100 thousand gallons of milk are demanded per week. b. Use this model to determine the amount of milk sold per week when the average price per gallon is $2.90. Solution In this part, a price of P 2.90 is specified. We can substitute this value into the model and solve for Q to get the quantity. 2.90 0.0395Q 7.0675 4.1675 0.0395Q 4.1675 Q 0.0395 Subtract 7.0675 from both sides Divide both sides by -0.0395 This fraction is approximately 105.506 thousand gallons of milk or about 105,506 gallons of milk per week. 9 Example 2 Interpret the Slope of the Model The least squares linear model for the data in Table 1 is P 0.0395Q 7.0675 . Use this model to determine the rate at which the prices are changing with respect to quantity. Solution The rate at which the variables change with respect to each other is the slope of the model. Figure 7 – A graph of the model with the slope labeled as a ratio. The slope, -0.0395, is the coefficient of the independent variable Q. However, giving the number as the rate is not complete. A rate should include units. This allows you to understand what the rate means. In Figure 7, the slope of the model is labeled so that the ratio of the vertical change to the horizontal change is 0.0395 0.395 10 This tells us that for every 10,000 gallon increase in demand, the price per gallon drops .395 dollars or 39.5 cents. 10 We could also write the ratio as 0.0395 0.395 . 10 In this case, we would interpret this as a decrease of 10,000 gallons in demand leads to an increase in price of .395 dollars or 39.5 cents. In general, rates are interpreted in 1 unit increments of the independent variable. Writing the ratio as 0.0395 0.0395 1 means that for every 1 thousand gallon increase in demand, the price drops by 0.0395 dollars or 3.95 cents. We can state this easily by saying that the rate is -0.0395 dollars per thousand gallons. The units include the term “per” meaning that the units are a ratio of dollars to thousands of gallons. 11 Question 3: How good is the linear model? The linear model is not complete without an indication of how good the fit is to the data. We can examine the scatter plot with the model and data and get a qualitative idea of the fit, but this can be deceiving. Figure 8 – Two scatter plots of the data in Table 1. The model of each scatter plot is P = -0.0395Q + 7.0675, but the horizontal and vertical scales are different. Which linear model in Figure 8 appears to be a better fit? On the surface, you would probably say that the model on the right is a better fit. But in fact, both scatter plots depicts the exact same model with different scales. The vertical scale for the scatter plot on the right is larger and makes any gaps between the model and the data seem small. This makes the points appear to be closer to the line. In fact, the vertical distance between each data point and the line are exactly the same and there is no difference in the fit. To remedy this and other difficulties in determining goodness of fit, two indicators are used. The correlation coefficient and coefficient of determination are commonly used to compare the fit of regression models. 12 The correlation coefficient r is a number from -1 to 1 that indicates how well the linear model fits a set of data. If r is closer to 1, the relationship between the data is more linear. If r is closer to 0, the data are not linearly related. A positive correlation coefficient indicates that the data is positively correlated. For linear models with a positive correlation coefficient, the slope of the model will be positive. A negative correlation coefficient indicates that the data is negatively correlated. For linear models with a negative correlation coefficient, the slope of the model will be negative. Figure 9 – Three different sets of data and the corresponding linear models. The worst fit is in graph a with an absolute value of the correlation coefficient closest to 0. The best fit is graph c with an absolute value of the correlation coefficient closest to 1. Each model is decreasing so the correlation coefficient is negative. As r gets closer to 1, the data points get closer to the linear model. Another measure of fit is the coefficient of determination, r 2 . For a linear model, the coefficient of determination is the square of the correlation coefficient r . Since r is a number from -1 to 1, r 2 is a number from 0 to 1. The closer the coefficient of 13 determination is to 1, the more linearly related the data are. If the coefficient of determination is close to 0, the data are not linearly related. Figure 10 – A graphing calculator or Excel can be used to calculate the correlation coefficient or coefficient of determination. A graphing calculator (left) returns both values. Excel (right) can return the coefficient of determination (it is written as R2 instead of r2. Another measure of how close the data lie with respect to the linear model is the percent error. The percent error at any value of the independent variable is found by dividing the error, P Pˆ , by the data value P . We can find the percent error by calculating Percent Error P Pˆ P at each data value where P is the price data and P̂ is the model’s estimate of the price. At Q 115 , the percent error is Percent Error 3.23 2.5250 0.218 3.23 or approximately 21.8%. This means that as a proportion of the price, the price at Q 115 is 21.8% above the model’s estimate of the price. 14 Example 3 Find the Largest Percent Error For the model P 0.0395Q 7.0675 and the data in Table 3, Q P P̂ P Pˆ 85 3.89 3.7100 0.1800 95 2.78 3.3150 -0.5350 105 3.01 2.9200 0.0900 115 3.23 2.5250 0.7050 125 1.69 2.1300 -0.4400 Find the quantity that yields the largest percent error. Solution Add a column to the table and calculate the percent error at each quantity. Q P P̂ P Pˆ P Pˆ P 85 3.89 3.7100 0.1800 0.1800 0.046 3.89 95 2.78 3.3150 -0.5350 0.5350 0.192 2.78 105 3.01 2.9200 0.0900 0.0900 0.030 3.01 115 3.23 2.5250 0.7050 0.7050 0.218 3.23 125 1.69 2.1300 -0.4400 0.4400 0.260 1.69 The largest percent error is 26% and occurs at Q 125 . This price is 26% below the linear model. 15