LSP 120: Quantitative Reasoning and Technological Literacy Section 118 Özlem Elgün

Why are we here? Data: numbers with a context Cell: each data point is recorded in a cell Observation: each row of cells form an observation for a subject/individual Variable: any characteristic of an individual

Why Data? 1) Data beat anecdotes “Belief is no substitute for arithmetic.” Henry Spencer Data are more reliable than anecdotes, because they systematically describe an overall picture rather than focus on a few incidents.

Why Data? 2. Where the data come from is important. “Figures won’t lie, but liars will figure.” Gen. Charles H. Grosvenor ( ), Ohio Rep.

Microsoft Excel Open excel Collect data Calculate sum, count, average

What is a linear function? Most people would say it is a straight line or that it fits the equation y = mx + b. They are correct, but what is true about a function that when graphed yields a straight line? What is the relationship between the variables in a linear function?

a linear function is a relationship that has a fixed or constant rate of change.

Does the data represent a linear function? The first thing we want to do is be able to determine whether a table of values for 2 variables represents a linear function:

We could just use a calculator to find the rate of change but using Excel is more efficient. Excel is a spreadsheet program in which each cell is able to save information and do various calculations. To enter a calculation, first enter "=" then either the calculation or the formula. Entering a formula using cell references allows you to repeat a certain calculation down a column or across a row. To determine if a relationship is linear in Excel, add a column in which you have calculated the rate of change and filling that columns. You must translate the definition of change in y over change is x to formula using cell references. ABC 1xyRate of Change =(B3-B2)/(A3-A2)

Note that we entered the formula for rate of change not next to the first set of values but next to the second. This is because we are finding the change from the first to the second. Then fill the column and check whether the values are constant. To fill a column, either put the cursor on the corner of the cell with the formula and double click or (if the column is not unbroken) put the cursor on the corner and click and drag down. If the rate of change values are constant then the relationship is a linear function. So this example does represent a linear function. The fact that the rate of change is 2.5 means that when the x value increases by 1, the y value increases by 2.5. ABC 1xyRate of Change

Deriving a Linear Equation Next step is to write the equation for this function. Let's refer back to the general equation for a linear function: y = mx + b. x and Y are the variables For the work you will do in this class, these letters will represent real variables not just generic letters. M is the slope or rate of change and b is the y-intercept or, algebraically speaking, the initial value. The initial value is the y value when x = 0. For this example, we just found that the rate of change is 2.5. Since there is no x value of 0 in the given table, we must calculate the initial value. Choose one set of x and y values and plug these and the rate of change into the general equation to find "b". Using the first set of values (x=3 and y=11) and 2.5 for "m": 11=2.5*3 + b. Solving: 11=7.5 + b ; 3.5 = b. The equation for this function is : y = 2.5 x Another way to find the equation is to use Excel. This will be demonstrated in class.

Practice For the following, determine whether the function is linear and if so, write the equation for the function. xy xy xy xy

Warning: Not all graphs that look like lines represent linear functions The graph of a linear function is a line. However, a graph of a function can look like a line even thought the function is not linear. Graph the following data where t is years and P is the population of Mexico (in millions): What does the graph look like? Now, calculate the rate of change for each set of data points (as we learned under Does the data represent a linear function?) Is it constant? tP

What if you were given the population for every ten years? Would the graph no longer appear to be linear? Graph the following data. Does this data (derived from the same equation as the table above) appear to be linear? Both of these tables represent an exponential model (which we will be discussing shortly). The important thing to note is that exponential data can appear to be linear depending on how many data points are graphed. The only way to determine if a data set is linear is to calculate the rate of change (slope) and verify that it is constant. tP

"Real world" example of a linear function: Studies of the metabolism of alcohol consistently show that blood alcohol content (BAC), after rising rapidly after ingesting alcohol, declines linearly. For example, in one study, BAC in a fasting person rose to about % after a single drink. After an hour the level had dropped to %. Assuming that BAC continues to decline linearly (meaning at a constant rate of change), approximately when will BAC drop to 0.002%? In order to answer the question, you must express the relationship as an equation and then use to equation. First, define the variables in the function and create a table in excel. The two variables are time and BAC. Calculate the rate of change. TimeBAC % %

TimeBAC Rate of change % %-0.008% This rate of change means when the time increases by 1, the BAC decreases (since rate of change is negative) by.008. In other words, the BAC % is decreasing.008 every hour. Since we are told that BAC declines linearly, we can assume that figure stays constant. Now write the equation with Y representing BAC and X the time in hours. Y = -.008x This equation can be used to make predictions. The question is "when will the BAC reach.002%?" Plug in.002 for Y and solve for X..002 = -.008x = -.008x x = 2 Therefore the BAC will reach.002% after 2 hours.