Linear Modeling Exercises. In case you d like to see why the best fit line is also called a least squares regression line here ya go!

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1 Linear Modeling Exercises Pages Problems 1 4, 5-9 (you might want to do the E exercises next), 20 In case you d like to see why the best fit line is also called a least squares regression line here ya go! Look back to exercise 9 (in the Larson book) and its data. Assume, for simplicity, that the tree is perfectly circular (although this argument holds even if it isn t). E1. Notice that your dependent variable is circumference. What are the units of circumference? E2. Estimate this tree s circumference in E3. What I d like to do is analyze the cross sectional area of the tree (that is, the area of the circle that would be left at the top of the stump if you cut the tree down). Using the following formulas, show me why the area of a circle is equal to the square of its circumference divided by 4 ; that is, Acircle = C Formula for circumference of a circle: Ccircle = d 2. Formula for the area of a circle: Acircle = d2 4 E4. What are the units of area, if circumference is measured in inches? E1. Inches. E Answers. E inches, according to my best fit line from problem 9. Don t worry about calculating a margin of error right now we ll get into those in MTH 243 and MTH 244! E3. Here are some hints: First, solve the circumference formula, Ccircle = d, for d. Next, substitute the result into the d in the area formula Acircle = d2 4. Cancel common factors, and you ve got it! E4. Square inches (or in 2 ).

2 Linear Modeling Quizzes Quiz 1. Year Number of Words The data at left shows the number of words in Facebook s privacy policy in recent years (pretty ironic for a site that prides itself on small snippets of the English language, eh?). What s interesting, to me at least, is that these data are pretty linear! For these data, use your Excel sheet to create the best fit equation of a line. Let x be years, and let y be the number of words. If you don t want to deal with scientific notation in the regression equation, let your x values be 5, 6, 7, and so on. 1. (2 points) Interpret the slope of this model, in context. 2. (2 points) According to this model, how many words should have been in Facebook s privacy policy in 2012? Note: don t worry about the MOE for this prediction. 3. (2 points) If the pattern continues, in which year will Facebook have more than 15,000 words in its privacy policy? 4. (1 point) Which of the questions above asked you to extrapolate? 5. (3 points) Name one reason that extrapolating with any data set can lead to inaccurate conclusions 1. 1 Right now, FB is dealing with blowback WRT possible interference in the 2018 US elections. As a result, its privacy policy is now a series of dozens of individual pages, containing what seems to be tens of thousands of words thereby proving the point that past performance doesn t guarantee future results.

3 Quiz 2. So, in class we did a couple of cost analyses (light bulbs, water heaters). I d like to do one more! A few years ago, my family decided it was time to put PV (photovoltaic) panels on our house. It was a big decision for us, since it required us to save for years before we actually did it. But, the time was right, the ETO, the state, and the feds gave us some good incentives, and we decided, Let s do this! So, with all of the incentives accounted for, we had to pony up around $5000 to cover the startup cost of the panels purchase and installation (technically, we had to provide even more, but with the tax breaks, it washed out to an overall out-of-pocket cost of $5000). Now, electricity in Bend carries an average rate of about cents (that s $0.1078) per kilowatt hour (a kilowatt hour, or kwh, is defined to be using 1000 watts of power for the duration of 1 hour ). Based on my own analysis of our power bills, we save (on average) around 15% on our electric bills by using the panels. 1. (2 points) What s the Rule family average electric cost per kwh? Looking at our power bill, we use about 11,000 kwh of power each year. How much does that power cost us? Do it two ways: 2. (2 points) As if we didn t have the panels (that is, use the $ per kwh) 3. (2 points) Using the panels (that is, use the rate you found in part 1). You now have two slopes and two y intercepts (technically, one s 0) to use in a cost analysis! Use your Excel sheet (the Exact tab) to approximate how many years it will take until our investment in solar will start saving us money. 4. (2 points) We installed the panels in In approximately which year will having the panels cost us less than not having them? 5. (2 points) Take a screen capture of the graph you used and include it with this quiz! Here s a video to help you take a screenshot if you need it! (link:

4 Quiz 3. This is a little quiz to extend our discussion from class about light bulbs (INC vs. CFL vs. LED). Some of you (very much correctly!) noticed that there was another variable to be considered lifespan of the bulbs! Here s the graphic we used to analyze these costs (one of the graphics, actually.we used a couple!): Use the above graph to answer the following! Approximate the number of hours; no need for exact values. 1. (2 points) Between hours and hours, the INC is cheaper than the CFL. 2. (2 points) Between hours and hours, the CFL is cheaper than the LED. But then, some of your awesome classmates noted that these bulbs need to be replaced every so often! In particular, INC bulbs last about 1200 hours, CFL about 8000, and LED around hours (on average)! So, accompanying this quiz is a spreadsheet where I changed up the one we used in class to allow for replacement costs. Open up that new sheet and check out the difference! So, in those spaces, you d need to place the number of hours after which you d have to replaces the bulbs, on average! Go ahead and fill in the Slope and Y-Intercept boxes like we did in class (in case you forgot them, they re in the PowerPoint for the Linear Modeling topic), and fill in the Replace After numbers with the ones above!

5 Now, mess around with the XMax value so you can see what the new coding does! See that litte hop up on the INC graph? That s the jump in price from having to pay an additional 50 cents for the next INC bulb! Cool! This window goes to 2000 hours let s go out to 5000 so you can see a few more jumps! OK! So, this is pretty cool we ve taken care of one more variable, just by coding the sheet a little differently. Sa-weet! 2 Next, change the XMax to (this represents the cost of running the bulbs for the maximum lifespan of the LED, the longest-lasting bulb). 3. (1 point) How many times would you have had to replace the INC, on average, in hours? 4. (1 point) How many times would you have had to replace the CFL, on average, in hours? 5. (2 point) (w) Based on what we talked about in class, about how many years is hours of light bulb use? We assumed, if I remember correctly, about 6 hours of bulb usage per day. So, if we average 6 hours of bulb usage daily, how many years will hours be? Remember that explaining how you used technology to get this answer is work! 6. (1 point) Roughly how much more, in dollars, would it cost to run an INC over hours than an LED? 7. (1 point) My family s house has about 40 light bulbs in it that could be replaced with LEDs. If they re currently all INCs, and we swapped all of them out for LEDs, how much money would we save, over hours? Heck yeah! I love math. 2 It was badass fun to code, too. If you want 3 extra credit points, figure out the formula I used to do it! Hint: you can click around in the sheet and look for it.

6 Quiz 4. This is a little quiz to extend our discussion from class about purchasing a 2012 Ford Escape should we purchase one with a hybrid engine, or a gasoline engine model? In class, we had different answers but all of them were based on costs per mile. We ll continue that here, with a few other variables taken into consideration! 1. (3 points) (w) If gas costs $2.50 per gallon, how many miles need to be driven until it becomes cheaper to run the hybrid than the V-6 (city driving)? By (w), I just mean show me what you needed to do to get your answer (I m thinking at least some conversions, and some graphing in the Excel Calculator. Feel free to take screen shots!). You ll need this info from class! 2. (1 point) What if gas costs $3.50 a gallon? No work needed for this or the next one I figure you ll just repeat what you ve already done. 3. (1 point) What about $4.50 a gallon? This is about how much gas averaged for a stretch back in 2008 and Americans reacted. Because of your answer to the last question, you might see why so many folks went out and bought hybrids in Of course, there are a couple of things that need to be considered: The price of gas didn t stay that high, that long; The cost of maintaining a hybrid car is, generally, higher than that of maintaining a gasoline car...but only because of the batteries! That s right if the batteries die in your hybrid, it s a lot of money. How much? Well, like so much in life, that depends. Doing some research, it looks like the cost of replacement battery packs has gone down over time, from close to $8000 years ago to more like $2000 nowadays (and this, of course, can vary widely for labor cost reasons). One more thing to be considered how many miles will these batteries work for? Again it varies! I read stories of hybrid taxis in NYC having batteries that lasted 300,000 miles and then heartbreak stories of clunker hybrids whose batteries die after 70,000 miles. So, let s let stuff vary! Go ahead and open up the spreadsheet that accompanies this quiz! (it s similar to the one from last week s quiz, if you did that one) 4. (4 points) Suppose that, as is often claimed, hybrid battery packs last for miles, and cost about $4000 to replace (this includes the parts and labor). Change those constraints in the spreadsheet. Also, use $3/gallon to get your cost per mile (that s what we used in class), and then adjust your Xmax to see what happens when you need to replace those batteries! Take a screenshot, include it here, and describe it in a few sentences here! Knowing this, some purchasers of used hybrids attempt to negotiate a lower purchase price to offset any of this potential goofiness! 5. (1 point) About what purchase price would you have to negotiate to remove the possibility that you saw in #4? Nearest 500 dollars rounding is fine for this!