lsp 120: quantitative reasoning and technological literacy

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LSP 120: Quantitative Reasoning and Technological Literacy. Topic 1: Introduction to Quantitative Reasoning and Linear Models. Foundations. 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 - PowerPoint PPT Presentation

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LSP 120: Quantitative Reasoning and Technological Literacy Section 118

LSP 120: Quantitative Reasoning and Technological Literacy

Topic 1: Introduction to Quantitative Reasoning and Linear ModelsPrepared by Ozlem Elgun1

Prepared by Ozlem Elgun2FoundationsData: numbers with a contextCell: each data point is recorded in a cellObservation: each row of cells form an observation for a subject/individualVariable: any characteristic of an individual

Prepared by Ozlem Elgun3Why Data?1) Data beat anecdotesBelief is no substitute for arithmetic.Henry SpencerData are more reliable than anecdotes, because they systematically describe an overall picture rather than focus on a few incidents .

Prepared by Ozlem Elgun4Why Data?2. Where the data come from is important.Figures wont lie, but liars will figure.Gen. Charles H. Grosvenor (1833-1917), Ohio Rep.

Prepared by Ozlem Elgun5

Prepared by Ozlem Elgun6Familiarizing with DataOpen Excel

Collect data:Ask 5 classmates the approximate # of text messages they send per dayRecord the data on Excel spreadsheet

Calculate average using the Average function on Excel. (There are many functions such as sum, count, slope, intercept etc. that we will use in this class)

Prepared by Ozlem Elgun7What 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 indicates a relationship between x and y that has a fixed or constant rate of change.

Prepared by Ozlem Elgun8Is the relationship between x and y is linear?

The first thing we want to do is be able to determine whether a table of values for 2 variables represents a linear function. In order to do that we use the formula below:

Prepared by Ozlem Elgun9To determine if a relationship is linear in Excel, add a column in which you calculate the rate of change. You must translate the definition of change in y over change is x to a formula using cell references. Entering a formula using cell references allows you to repeat a certain calculation down a column or across a row. Once you enter the formula, you can drag it down to apply it to subsequent cells. ABC1xyRate of Change23113516=(B3-B2)/(A3-A2)4721592661131

This is a cell referencePrepared by Ozlem Elgun10Note 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. Rate of change is 2.5 and it is constant. This means that that when the x value increases by 1, the y value increases by 2.5.

ABC1xyRate of Change231135162.547212.559262.5611312.5Prepared by Ozlem Elgun11How to Write a Linear EquationNext step is to write the equation for this function.

y = mx + b. y and x are the variables m is the slope (rate of change) b is the y-intercept (the initial value when x=0)

We know x, y, and m, we need to calculate b:Using the first set of values (x=3 and y=11) and 2.5 for "m (slope):

11=2.5*3 + b. Solving: 11=7.5 + b 3.5 = b.

The equation for this function is : y = 2.5 x + 3.5

Another way to find the equation is to use Excels intercept function.

ABC1xyRate of Change231135162.547212.559262.5611312.5Prepared by Ozlem Elgun12Next step is to write the equation for this function.

y = mx + b. y and x are the variables m is the slope (rate of change) b is the y-intercept (the initial value when x=0)

We know x, y, and m, we need to calculate b:Using the first set of values (x=3 and y=11) and 2.5 for "m (slope):

11=2.5*3 + b. Solving: 11=7.5 + b 3.5 = b.

The equation for this function is : y = 2.5 x + 3.5

Another way to find the equation is to use Excels intercept function.

ABC1xyRate of Change231135162.547212.559262.5611312.5Practice: For the following, determine whether the function is linear and if so, write the equation for the function.

On the class website- click on Rate of change calculations.xlsx

xy5-410-1152205xy112359718xy220413668-1Prepared by Ozlem Elgun13"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 0.018 % after a single drink. After an hour the level had dropped to 0.010 %. 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.

TimeBAC00.018%10.010%Prepared by Ozlem Elgun14TimeBACRate of change00.018%10.010%-0.008%2This 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 + .018.

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 + .018-.016 = -.008xx = 2 Therefore the BAC will reach .002% after 2 hours.Prepared by Ozlem Elgun15

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?tP198067.38198169.13198270.93198372.77198474.67198576.61198678.60Prepared by Ozlem Elgun16What 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.

tP198067.38199087.102000112.582010145.532020188.122030243.162040314.32Prepared by Ozlem Elgun17Linear Modeling-TrendlinesThe Problem - To date, we have studied linear equations (models) where the data is perfectly linear. By using the slope-intercept formula, we derived linear equation/models. In the real world most data is not perfectly linear. How do we handle this type of data? The Solution - We use trendlines (also known as line of best fit and least squares line).

Why - If we find a trendline that is a good fit, we can use the equation to make predictions. Generally we predict into the future (and occasionally into the past) which is called extrapolation. Constructing points between existing points is referred to as interpolation.

Prepared by Ozlem Elgun18

Is the trendline a good fit for the data?

There are five guidelines to answer this question:

Guideline 1: Do you have at least 7 data points?

Guideline 2: Does the R-squared value indicate a relationship?

Guideline 3: Verify that your trendline fits the shape of your graph.

Guideline 4: Look for outliers.

Guideline 5: Practical Knowledge, Common Sense

Prepared by Ozlem Elgun19

Guideline 1: Do you have at least 7 data points?

For the datasets that we use in this class, you should use at least 7 of the most recent data points available. If there are more data points, you will also want to include them (unless your data fails one of the guidelines below). Prepared by Ozlem Elgun20Guideline 2: Does the R-squared value indicate a relationship?R2 is a standard measure of how well the line fits the data. (Tells us how linear the relationship between x and y is)

In statistical terms, R2 is the percentage of variance of y that is explained by our trendline.

It is more useful in the negative sense: if R2 is very low, it tells us the model is not very good and probably shouldn't be used.

If R2 is high, we should also look at other guidelines to determine whether our trendline is a good fit for the data, and whether we can have confidence in our predictions.

Prepared by Ozlem Elgun21More on R-squaredIf the R2 = 1, then there is a perfect match between the line and the data points.

If the R2 = 0, then there is no relationship between n the x and y values.

If the R2 value is between .7 and 1.0, there is a strong linear relationship and if the data meets all the other guidelines, you can use it to make predictions.

If