What Makes a Track Fast?

Transcription

1 Tracks on the NASCAR circuit are measured using multiple attributes. Statistics can range from length of track, banking degrees in curves, average speed of laps, width of lanes, to many other features. When sifting through all of those numbers, how do we decide which tracks are truly the fastest? Let s use the following data: Track Name Length of track (in miles) Banking (in degrees) Fastest Lap in 2010 (mph) 1 Atlanta Motor Speedway Auto Club Speedway (Fontana, CA) Bristol Motor Speedway Charlotte Motor Speedway Chicagoland Speedway Darlington Raceway Daytona International Speedway Dover International Speedway Homestead- Miami Speedway Indianapolis Motor Speedway Kansas Speedway Las Vegas Motor Speedway Martinsville Speedway Michigan International Speedway New Hampshire Motor Speedway Phoenix International Speedway Pocono Raceway Richmond International Raceway Talladega Superspeedway Texas Motor Speedway This data has three components: Length of Track: This is the total length of one revolution around the track Banking: This is the degree of banking in the steepest portion of the track Fastest Lap in 2010: This is the fastest speed of the driver during their qualifying laps that landed the pole position at one of the races held at the track in the 2010 season. If there was racing at the same track twice during 2010, then the fastest lap was chosen. Page 1

2 Exploration Questions: 1. From the data given, which track appears to be the fastest and why? Which track appears to be the slowest and why? 2. A confounding variable is an outside variable beyond the control of the race teams and officials. What are some confounding variables that could contribute to the speed of the tracks? 3. Describe in your own words how the track s banking affect the speed of the cars. 4. Compare the statistics for Martinsville Speedway and Pocono Speedway. Why is the fastest speed for Martinsville lower than Pocono even though Pocono has lower banking? Page 2

3 w e r t y u i o p [ ] \ a s dbuilding f SPEED g h j k l ; z xuse your c Graphing v Calculator b nand follow m the, directions. given / to answer the next questions: Extended Characters acters are accessed Enter by holding the data down for the the Alt tracks key into then your typing calculator the numbers using on the numeric following key instructions: pad.) a. Press, and choose 1: Edit. Be sure to clear out any existing lists. b. Under d, enter the track number c. Under e, enter the track length d. Under f, enter the track banking e. Under g, enter the track speed Next, we want to determine if there is a correlation between the banking of the track and the maximum speed of the track. Using the following instructions, graph the scatterplot of this data: a. Press y, o,, (this should highlight the on for Plot 1) b. Scroll down to highlight the selection for the XList and change it to f (which is our banking data) by pressing y, 3. c. Scroll down to highlight the selection for the YList and change it to g (which is our speed data) by pressing y, 4. d. Press q, Graph the scatterplot of your data below. Be sure to label each axis. Page 3

4 w e r t y u i o p [ ] \ Building SPEED a s d f g h j k l ; 6. Does there appear to be a correlation between the banking degrees and the z x speed cof the vtrack? bwhy or n why not? m If yes,, is the. correlation / positive or negative? Extended Characters aracters are accessed by holding down the Alt key then typing the numbers on the numeric key pad.) Now, let s determine a line of best fit for the data by following these directions: a. Press, right arrow to /, 4:LinReg(ax+b) b. The calc should now say LinReg(ax+b) and you need to tell it to compute the statistics based on f and g by pressing y, 3,, y, 4,. 7. What is the line of best fit for the banking and speed data? 8. There are many different guidelines for the interpretation of a correlation coefficient. It has been observed, however, that all such criteria are in some ways arbitrary and should not be observed too strictly. This is because the interpretation of a correlation coefficient depends on the context and purposes. What is a strong correlation for one situation may be weak in another situation. For example, a correlation of.9 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences where there may be a great contribution from complicating factors. Correlation Negative Positive Use the chart to the right to interpret the correlation for this set of data. What is the correlation Small 0.3 to to 0.3 coefficient (r-value) for this data? Do you think this Medium 0.5 to to 0.5 is a strong, moderate, or weak correlation? Describe Large 1.0 to to 1.0 the relationship between these two numbers. (you may need to turn Diagnostics on) Page 4

5 9. Using the information you have stored in your calculator and what you know about correlation coefficients, try to find a stronger relationship between two other variables. Are there two variables with a stronger correlation than banking and speed? If yes, what are they? 10. Discuss why you think the results of the question above had that outcome. Page 5

6 EXTENSION: 11. Use your calculator to sketch the scatterplot comparing e (length) and g (speed). By observing the graph, do you still think a linear regression is the best fit for this model? Why or why not? 12. Try computing Quadratic Regression on the data to determine the quadratic equation that may best describe the data. Don t forget to manually enter e, g after your regression input. What is the quadratic equation that best models the graph? 13. What is the new r- value of this model? Is this a better fit than the linear regression found above? 14. Kentucky Motor Speedway is considering bring a NASCAR Sprint Cup race to their 1.5 mile track. Using your quadratic equation, estimate the speed the cars would be able to travel at this new race. Page 6

7 ANSWERS: Exploration: 1. Fastest = Texas, Slowest = Martinsville, They had the highest and lowest qualifying records, respectively. 2. Confounding variables = weather (temp, precip, cloudcover), track conditions (surface type, wearing, etc) 3. It helps the cars keep momentum and speed in curves. 4. Martinsville is a shorter track and doesn t have long straightaways for a greater chance at reaching a higher speed. Calculator: 5. Graph varies 6. There could be a slight positive correlation (students could answer with no correlation because it is minimal) 7. y = 1.30x r = 0.336, there is a medium positive correlation 9. Track length and speed have a stronger correlation at r = Those tracks have longer straightaways are more opportunities for gaining higher speeds. Extension: 11. No, it appears to have a curved shape 12. y = x x R 2 = 0.869, so R=.932 much better fit! mph (For the inaugural race, qualifying was rained out, so we will have to check this speed next summer!) Page 7