Quadratic Regressions Group Acitivity 2 Business Project Week #4

Transcription

1 Quadratic Regressions Group Acitivity 2 Business Project Week #4 In activity 1 we created a scatter plot on the calculator using a table of values that were given. Some of you were able to create a linear regression using your calculator s linear regression application. In this activity, we will look at data that is best fit by a quadratic regression. Remember that a quadratic is in the shape of a parabola. The parabola can either open up or open down. We will create a scatterplot by hand, create a scatterplot on our calculator, and find a regression equation on our calculator. Remember, that a regression equation is an equation that best fits a given set of data. It best describes the relationship between the two variables. The regression equation follows the trend of the data. Getting Started with a Data Set Profit In manufacturing, profit is often modeled using a quadratic equation. For this instance, assume we are manufacturing cell phones. Let n be the number of cell phones manufactured each month. If a month exists where we manufacture zero cell phones, we still have expenses to be paid. These expenses are called fixed costs. This implies that the company will lose money and the profits will be negative when zero cell phones are manufactured. As we begin to manufacture and sell cell phones, we will start to bring in revenue. At first the company will be working at a loss. At some point, the company will create enough cell phones that the company will start to make a profit. This point where loss turns to profit (or profit turns to loss) is call the break even point. As the company makes more and more cell phones, they will need to sell the phones for lower and lower prices. It is easy to see that when the sale price of the phone becomes less than the cost of manufacturing the phone, the company will again have negative profits. Because of fixed costs, this actually occurs long before the sale of the cell phone becomes less than the manufacturing cost. The following data estimates the monthly profit for a cell phone manufacturing company based on the number of cell phones they manufacture each month. n (Number of Cell Phones) P (Profit or loss) $35,000 -$18,000 -$2,000 $51,000 $4,900 -$3,900

2 Plotting the Data by Hand You will be plotting the data on the grid shown below, graphing the number of cell phones on the x-axis (number of cell phones is the domain and the independent variable). You will be graphing profit on the y-axis (profit is the range and the dependent variable). Remember that the domain is the set of all possible values of an independent variable of a function and the range is the set of all possible values for a dependent variable of a function. These are the same steps that we followed in activity 1. Step 1: Label the n-axis (normally this is the x-axis but we are using n to represent number of cell phones) and the P-axis (normally this is the y-axis but we are using P to represent profit) of your grid. Let s start with the n-axis. What is your smallest n-value? What should be your smallest n-value on your n-axis? (Make sure that your smallest value is on the n-axis. You need to have your n-axis go beyond the smallest value.) What is your largest n-value? What should be your largest n-value on your n-axis? (Make sure that your largest value is on the n-axis. You need to have your n-axis go beyond the largest value.) Label these values on your grid below. What is your smallest P-value (Profit value)? Think about this, can your height be negative? What should be your smallest P-value on your graph? (Make sure that your smallest P- value is on the P-axis. You need to have your P-axis go beyond the smallest value.) What is your largest P-value (Profit value)? What should be your largest P-value on your P-axis? (Make sure that your largest value is on the P-axis. You need to have your P-axis go beyond the largest value.) Label these values on your grid below. Step 2: Now that your grid is labeled on the n and P-axis, plot the data points from the table onto the grid.

3 Step 3: Look at your data. If you were to find an equation of a function that best fits this data, would this equation be a line? Why or why not? Would this equation be a parabola (quadratic)? Why or why not? **Make sure to remember that you are trying to find the equation that would best fit the graph. Not the equation that exactly fits the graph. If you have plotted the data correctly the equation that will best fit this data is a parabola (quadratic). Sketch a parabola that best fits this data. Explain why your sketch of a parabola best fits the data. Finding the Equation for the Quadratic of Best Fit Using Your Calculator Now, you will enter this data into the calculator and use the calculator to find the equation of your quadratic of best fit. Please use the instructions on the document labeled Regression Instructions on the Math 143 MLC webpage. First, find the set of instructions that says Graphing a set of data points on the TI-83 (making a scatterplot). Follow these instructions to plot the points on your calculator. Once you have plotted the data on your calculator and you see a plot similar to the one that you sketched, observe the scatterplot. Does your graph on the calculator look like the graph that you sketched? Why or why not? If they do not look the same you might want to explore reasons such as window size, domains and ranges, sketched points correctly or entered data correctly in the calculator.

4 Once you have resolved any differences between the sketch and the calculator, look at the graph on your calculator. Do you think that a line best fits the data? Why or why not? Do you think that a quadratic best fits this data? Why or why not? If the data was entered and plotted correctly, it should most resemble a quadratic just like in your sketched graph. Now you want to find a quadratic that is a good "fit." We will find this equation using your calculator. In the document labeled Regressions, find the section labeled Finding a Quadratic Regression. Use those instructions to find the equation for the quadratic that best fits the data. What is the quadratic equation that best fits the data? After you have found a quadratic equation to best fit the data, try to find a cubic equation to fit the data. Follow the steps that you did for the quadratic regression but instead use the CubicReg key. What is the cubic equation that best fits the data? Graph this cubic equation in your calculator. Does this graph do a good job of approximating the data? Yes or No. If you answered the last question, No, the cubic regression does not fit the data points Move on. If you answered yes, think about a cubic regression. A cubic regression should approximate data points better than a quadratic regression. GO back and determine what you did wrong. Explain what you did wrong and fix the issue. Hint: Look at the coefficients for the regression very carefully. What does the E on your calculator represent? Notice that the cubic regression and the quadratic regression both fit the data points, what are the similarities (differences) between the two?

5 Pick either the cubic or the quadratic regression to model the data set. Explain why you chose that equation? (Hint: pick the quadratic why?) Finally, we can use our model to make predictions or find other points on the curve that were not given. Using your model, after 400 cell phones are manufactured, what is the profit? Explain what the signed number means for the company. Locate this point on both your stetched graph and the graph on your calculator. Label this point on your sketched graph. What are the Break-Even points? What is the maximum Profit this model allows for? How many cell phones should you attempt to manufacture for maximum profit? What is the number of cell phones you should manufacture in order to remain pofitable? Note: This answer should be a range of numbers why? What does the model suggest that the amount of fixed cost is?

6 Exercises For both of the data sets below: a. determine the domain and range, b. plot the data on graph paper making sure to label your window on your paper, c. sketch a quadratic of best fit, d. enter the data into your calculator, e. find the quadratic of best fit. 1. For the data below, determine the break-even points and explain what number of cell phones manufactured will create a profit. n (Number of Cell Phones) P (Profit or loss) $20,000 $48,000 $52,000 -$31,000 -$950, For the data below, find the maximum profit and the number of cell phones manufactured to produce the maximum profit. Also find the estimated fixed costs. n (Number of Cell Phones) P (Profit or loss) $18,000 -$10,000 -$4,000 $250 $2000 $1200