Math 1525: Lab 4 Spring 2002

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1 Math 1525: Lab 4 Spring 2 Modeling---Best Fit Function: In this lab we will see how to use Excel to find a "best-fit equation" or model for your data. Example: When a new motion picture comes out, some fans rush to see the movie during the first weekend of release. Casual movie-goers, however, often wait to see if the reviews are good, or until the movie comes out at a discount chain. That means for some movies, there is a lull between the first wave of fans and the second wave of casual viewers. Here is the attendance data for such a movie: Let s model this data. First we should graph the data points to get a clear picture of what the function will look like. week attendance (thousands) The plot above is the graph of the data using 'xy-scatter' in Excel. Notice that we use the data points graph only and do not connect these points this time. This is because no matter what our finished model might look like, we re ultimately dealing with discrete, separated points. To choose a model you must first look at the shape of your data points. For this particular data, if you were to describe the overall shape of the graph, you might say: The graph starts out high, but drops off quickly. It peaks again after a while, then drops off again. We will now try to produce an equation that will come the closest to representing this data and the TREND of this data. First single click on the chart. Next click on any one of the data points. All data points will be highlighted. Click on Chart and then add Trendline. A Dialogue box will appear and then you click on Type and choose the type of equation that you wish to view over your data. You may wish to try a linear model first. Highlight Linear box and then click on OK to view the linear model. Click off the chart to see the curve better. VA Tech Department of Mathematics 1

2 Notice that the linear model is not a close fit but it may or may not give a good sense of the TREND of the data. Now try a different model. Highlight the trendline on your chart and select Edit-Undo Trendline. The linear trendline should be gone and we will try another one. If you have a problem, click on an empty cell elsewhere in the spreadsheet and then go back to the chart Notice that this is a set of data points that change directions twice (down, then up, then down again); this suggests that a cubic trendline might fit this model best. Let s add a cubic trendline to the graph and see what we get. Highlight polynomial, then click on order 2 and change it to order 3. Go to Options, select Display Equation on Chart, and then click on OK. If you closed the dialogue box before selecting the option to display the equation, single click on the trendline and click on Chart- add Trendline-Options, etc. This fit is much better. To change the appearance of your trendline, highlight the trendline and click on Format. You will have a variety of patterns, type, and options. Overall, this isn t a bad fit. The shape of the trendline roughly matches the shape of the points. 1 1 y = x x x Look closely at the point (4, 7). Your instincts might tell you that this point seems really out of place with the rest of the data. And your instincts would be right. We don t necessarily know what s wrong with the point. Maybe there s a good reason why that week had unusually low attendance (for example, the theater is in a college town, and week 4 was spring break), or maybe there s no good reason, and that 7 was just a typo. We don t know, but as business professionals, we have the prerogative to occasionally omit data points that we think are suspect. Doing so can often improve the quality of the overall model. VA Tech Department of Mathematics 2

3 On the right is a graph of the same data with our bad point removed. Notice that the equation of the trendline is different and it matches the overall shape of our data points even more closely than before! If you choose to omit some of your data when developing a model, it s a good idea to inform your client that you have done so and explain the reasons why. 1 1 y = x x x Finally, we re ready to complete our model. Choose some descriptive variables, and remember to explain what they mean. Here s one way you could write the model: Equation: A(w) = w w w Definition of variables: w = number of weeks since the movie opened A = approximate attendance (in thousands of people) during week w. Data for week 4 was not included due to exceptionally low attendance. Do you think Attendance will indeed continue to decline for this movie? Note: When you are in the Trendline option box you will notice a box option for a logarithmic function. If that box is blank or it will not work when you click on it, then you have an independent data point of zero included in your data set and log functions are not defined for zero. If you go back and delete that value from your database then you can make use of the log function options box. Steps in Choosing a Model: 1. When you choose a model or "best fit function" for a set of data you are NOT looking for a function that fits ALL the data points. To begin with, this is usually not possible and not necessary. Find a model that fits the TREND of the data set. For instance, if the data seems to be increasing overall then may be a linear equation with a positive slope or a quadratic equation that turns up or maybe an exponential equation will be what is needed. 1. The model you choose usually depends on the questions you need answered. If you need to find the maximum value only, then you just need to zoom in on a few of the points and disregard the rest. Then you only need a model to fit those few points that you are interested in. 1. Keep your models simple. Do not try to use a sixth degree equation when a quadratic will do just as well or better. 1. Look at the curvature of the scatter plot of the data. a) If the points appear to be in a straight line, try a linear model. a) If the scatter plot is curved but has no inflection point, try a quadratic and or exponential. VA Tech Department of Mathematics 3

4 a) If the scatter plot appears to have an inflection point, try a cubic and or logistic model ( Unfortunately Excel will not do a Logistic model) a) If the scatter plot indicates curvature not easily modeled by the above suggestions then consider combining two or more models to form a piecewise continuous model. 1. You need to narrow the choices to two models and choose the one that best satisfies the trend of the data set. (Usually the one that comes closest to all points but does not necessarily go through the most points is the best choice of the two but not always.) 1. If you can not see a 'best' choice between the two model choices then consider the answers that extrapolation would give. Which answers make the most sense in light of the problem scenario? 1. Sometimes either model may be a good choice. For instance if you wish to find the maximum point, then a quadratic equation that turns down or a cubic equation might both answer that question with in certain conditions. THINGS TO REMEMBER: 1. Always check to see if your data needs to be aligned. This will often give you equations that are more convenient to work with. 1. Interpolate: To use your equation to fill in data that may be missing within a given data set. 1. Extrapolate: To use your equation to predict the future trend of the given data set. 1. Quadratic means a polynomial equation of degree (order) 2, Cubic means a polynomial of order 3, and Quartic means a polynomial of order Hint on how to use trend line to find equations for piece-wise functions. VA Tech Department of Mathematics 4

5 Example: Consider the following table: x y This data should be divided at (4, 11). Go to Excel and type in the data as shown below and graph (See graph below.). Notice the data point groups will be in two different colors. Click on the first set of data and use trendline to find an equation and then use trendline again by clicking on the second data set and proceed as before to get the second equation. X Eq #1 Eq# y = x x + 13 Eq #1 Eq#2 Linear (Eq #1 Poly. (Eq#2) 1 y = 2x X VA Tech Department of Mathematics 5

6 Problems to be turned in: The following data sets can possibly be represented by a quadratic, cubic or exponential function. Use Excel to find the best-fit function for each data set. You will want to use the xy-scatter graph that is just dots. You answers should be typed neatly on your lab and in complete sentences. Label all graphs clearly and use the chart option to display the equation of the best-fit function found. 1. Mary made an initial deposit of $25 into an account that pays 6% per year compounded yearly: Year Balance $25. $265. $2.9 $ $ (a) Align your data before using Excel. (b) Do 3 graphs on 3 separate axes, one for each of the three function types: quadratic, cubic and exponential. Add trendline with equation on each graph. (c) Why is the exponential fit is the best choice here? 2. Spock and Kirk found a new life-form which they named Hokies. The data below shows how this population changes over different stages of time: Stage Pop (a) Find the exponential model by hand using y = ba t. Then convert the model to the form y = be kt. (b) Use trendline to develop an exponential model for the population. (c) Compare both equations found by hand and by trendline. Are they close? (d) Use your trendline model to predict (extrapolate) the Hokie population at stage Complete problem 32, parts (a), (b), (c), and (d) at the end of section 2.4 (page 154) in textbook. Be sure to include your graph with trendline as discussed in part b. Also don t forget to align your data so that 19 is year. 4. Complete problem 14 at the end of section 2.6 (page 174) in textbook. For the data set given in this problem you should do a scatter-plot of the data and examine your graph to determine the best model. Don t forget to align your data so that year 193 is year before graphing. Next, find the equation for the best-fit model to the data points using trendline. Then give the equation for your model, define the variables you use, include units of measure, and explain why you chose that model. In part a) you will discuss exponential growth, logistic growth, linear, quadratic, and cubic only. VA Tech Department of Mathematics 6

Math 2524: Activity 1 (Using Excel) Fall 2002

Math 2524: Activity 1 (Using Excel) Fall 22 Often in a problem situation you will be presented with discrete data rather than a function that gives you the resultant data. You will use Microsoft Excel