Statistics and the Common Core
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- Mean as Least Squares
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Slope InterpretationRate of Change in Context NAME____________________________ 1. To open a McDonald’s restaurant, assume it requires $1,500,000 in startup costs. The annual profit of a McDonald’s restaurant is around $250,000. a) Create a linear model to represent profit, P (in dollars) as a function of y (years since opening). b) Interpret the slope in context. c) Interpret the y-intercept in context. 2. On December 12, 2013, a weather balloon was launched from the grounds of Noblesville High School. There was an approximately linear association between the elevation above sea level of the balloon, E, (in feet) and the time from launch, t (in seconds). The least squares regression equation was 
a) Interpret the slope in context. b) Interpret the y-intercept in context. c) The elevation of the launch site is 784 feet above sea level. Does the linear model confirm or contradict this fact? Explain fully. 3. On December 12, 2013, a weather balloon was launched from the grounds of Noblesville High School. There was an approximately linear association between the longitude, L, in degrees and time from launch, m in minutes. The Prime Meridian in Greenwich, England has a longitude of 0˚. One way of measuring points west of the Prime Meridian are to denote them as negative degrees. All cities in the United States have negative longitudes. They range from the easternmost point in Sail Rock, Maine at –66.95˚ and the westernmost point in the Bodelteh Islands (offshore from Cape Alava, Washington) at –124.77˚.
b) Interpret the y-intercept in context. c) What general direction was the balloon travelling during launch? Explain.
How do you find the mean of five numbers? Add them up and divide by five, right? Another property of the mean is that it is the number that produces the least possible sum of the squares of the differences between each data point and this number. Complicated, huh? Let’s try it with the following data set: {1, 3, 4, 5, 9} 1. Assume you do not know the mean. What is a good guess? Let’s guess 5. Calculate all five differences from 5: _____ _____ _____ _____ _____ Now square all five differences: _____ _____ _____ _____ _____ Find the sum of the squares: _______ 2. Now do the same calculation with a guess of 4. Calculate all five differences from 4: _____ _____ _____ _____ _____ Now square all five differences: _____ _____ _____ _____ _____ Find the sum of the squares: _______ 3. Which of the guesses gave the least sum of squares? _______ But is this the number that gives the least POSSIBLE sum of squares? Maybe…maybe not… So let’s try some other tools: simulation and algebra! 4. There is a Fathom Demo that your teacher may show you here… 5. So in the Fathom Demo, the graph of all possible means vs. the sum of the squares using that mean is a parabola. So if we can find the vertex of this parabola, we can find the mean that produces the least sum of squares (which will be the mean).
So the parabola has the formula: S = _______________________________ 7. Now find the vertex using algebra or a graphing utility (or both!). Graphing on desmos.com is another option.
Signature (in cursive): _____________________________________________ Measure the following in centimeters:
1. Predict the shape of the sibling data. Explain your prediction. 2. What might we see when we graph a dotplot of height data? 3. Do you know what Leonardo DaVinci said about arm span and height? Google “Vitruvian man” and you will find out. How could we test this theory with our data? DaVinci also said that foot length is one-seventh of height. How could we investigate this claim? 4. Do people with longer signatures tend to also have taller signatures? Explain how you could find out. 5. Why would knowing average arm spans or heights or finger lengths (or knee heights) be important? 6. What would the shape of the “states visited” distribution likely be? Explain. PART 2: 7. Measure the objects provided by your teacher. Sketch a graph and describe each distribution. Object #1:_________________________________ Your measurement: _____________ Sketch and description: ______________________________________ Object #2: _________________________________ Your measurement: _____________
______________________________________ What could be the sources of the variation in the measurements?
9. Average speeds of the winning Kentucky Derby horses. What are some possible explanations for the possible outlier around 31.5 mph? 10. Finishing times of Kentucky Derby winning horses. What might explain the clear bimodal shape of this distribution?
11. Minutes played per game, all NBA players for one season. What would explain the different modes in this distribution? 12. Play 20 questions to see of you can guess what this distribution represents. 13. The first 999 Old Faithful eruption duration times of 2011 are graphed below. What unusual feature do you see in the histogram? Download 283.31 Kb. Share with your friends:
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(the Greek letter for m, pronounced “myoo”).