Welcome to LSP 120 Dr. Curt M. White

What is LSP 120? First course in QRTL (Quantitative Reasoning and Technological Literacy) Also known as Quantitative Reasoning First Year Program requirement Prereq to course is ISP 110 or Math 101 or placement through advising Know this stuff already? Take the exam this week.

What is LSP 120? We will examine data – look at it, talk about it, graph it, manipulate it To graph and manipulate we will use mathematics and technology Why do we want to do this?

What is LSP 120? Because the person that can use data (information) gains knowledge and thus has power! In college – writing research papers and taking advanced courses In your job – career advancements In government – getting things done (?) In life – more intelligent life decisions

What is LSP 120? “Information is a beacon, a cudgel, an olive branch, a deterrent – all depending on who wields it and how. Information is so powerful that the assumption of information, even if the information does not actually exist, can have a sobering effect.”

What are the topics? Linear and near-linear models (with trendlines and forecasting) Exponential models Useful percentages Graphing Consumer Price Index Absolute versus relative values Financial models

What is LSP 120? Let’s take a closer look at the syllabus

Linear Models and Trendlines LSP 120 Linear Models and Trendlines

Linear Relationships What makes a graph or function or table of values linear? (You have already seen this!) For a fixed change in x, there is a fixed change in y, or The change in y per unit change in x is constant, or There is a constant rate of change

Linear Relationships Why do we care if a set of data is linear? If a data set is linear (or near-linear), then we can better predict where the data will be in the future We can also go back in time and see where the data has been!

Linear Relationships Look at a set of values. Is it a linear relationship? Apply (B3-B2)/(A3-A2) Column A Column B (X values) (Y values) 2 4 25 3 5 50 4 6 75 5 7 100 6 8 125 7 9 150 No entry in this first row =(B3-B2)/(A3-A2) = (50-25)/(5-4) = 25 =(B4-B3)/(A4-A3) = (75-50)/(6-5) = 25 =(B5-B4)/(A5-A4) = (100-75)/(7-6) = 25 =(B6-B5)/(A6-A5) = (125-100)/(8-7) = 25 =(B7-B6)/(A7-A6) = (150-125)/(9-8) = 25 All the results are the same (25), so this is a linear set of values.

Linear Relationships If it is linear, what is the function? Recall: y = mx + b m is the slope, or the (change in y) / (change in x) b is the y intercept So calculate the slope Then plug in slope and first x and y values into y=mx+b and solve for b

Linear Relationships A(or X) B(or Y) 2 4 25 3 5 50 4 6 75 5 7 100 2 4 25 3 5 50 4 6 75 5 7 100 6 8 125 7 9 150 No calculation in this row (B3-B2)/(A3-A2) = (50-25)/(5-4) = 25 (B4-B3)/(A4-A3) = (75-50)/(6-5) = 25 (B5-B4)/(A5-A4) = (100-75)/(7-6) = 25 (B6-B5)/(A6-A5) = (125-100)/(8-7) = 25 (B7-B6)/(A7-A6) = (150-125)/(9-8) = 25 y = mx + b m = change in y / change in x = 25/1 = 25 25 = 25 * 4 + b b = -75 (this is the y-intercept) y = 25x – 75 This is the equation of the line

Examples x y 5 -4 10 -1 15 2 20 5 x y 0 1 2 4 4 16 6 36 x y 3 5 4 9 6 11 9 17 m=change y / change x = 3/5=0.6 y = mx+b -4=0.6*5+b b=-7 y = 0.6x - 7 x y Rate of Change 3 5 6 9 =(B3-B2)/(A3-A2) 9 13 12 17

Linear Relationships Linear growth – occurs when a quantity grows by the same absolute amount Exponential growth – occurs when a quantity grows by the same relative amount – that is by the same percentage – in each unit of time There is also linear decay and exponential decay

Examples The number of students at Wilson High School has increased by 50 in each of the past four years. The price of milk has been rising with inflation at 3% per year. Tax law allows you to depreciate the value of your equipment by $200 per year. The memory capacity of computer hard drives is doubling approximately every two years. The price of DVDs has been falling by about 25% per year.

Trendlines Real data is seldom perfectly linear Suppose the data is reasonably linear or near-linear? How does one make a linear model of the data? The standard approach is to use a “best-fit” line

In Excel – note the equation for the trendline and the R2 value.

Trendlines If R2 = 1, then 100% of the variance in y is explained by the line, so we have a perfect fit (the data is linear) If R2 = 0, then we have a terrible fit. (Better not make a prediction) What if R2 value = 0.5?

Trendlines Do we have to do this calculation? No, Excel can do it for you Let’s say you have made a graph of some data using Excel

After you make a graph, right-click on any datapoint. Select Add Trendline… Then check the two bottom boxes

Trendlines Let’s take a look at the dataset Motorcycles_By_Year2005.xls Graph the data using an XY Scatter. After the graph is done, right click on a data point, select Add Trendline

Trendlines How do you make a prediction? Two possible techniques: 1. Extend the trendline using Excel 2. Use the slope / intercept model (next slide) Be careful! One’s confidence in predictions made far from the data must be tempered!!

Trendlines You can use the equation for the trendline, but the terms in the trendline equation are significantly rounded. Can remove some of this rounding Right-click on the formula in the graph and then click on Format Trendline Label Select Number on left and right Let’s give it 6 decimal places

Trendlines You can also have Excel calculate the Slope and Intercept individually, and then use them in an equation Somewhere in Excel, enter label Slope and in next cell to right, enter =Slope( Do the same thing for Intercept

Let’s Go! Let’s head to the lab and start our first Activity But first, let’s break up into groups