Building SPEED What Makes a Track Fast?

What Makes a Track Fast?

Tracks on the NASCAR circuit are measured using multiple attributes. Statistics can range from length of track, banking degrees in curves, average 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 Banking Fastest Lap in (in miles) (in degrees) 2010 (mph) 1 Motor Speedway 1.54 24 192.761 2 (Fontana, CA) 2 14 185.285 3 .533 30 124.630 4 1.5 24 191.544 5 1.5 18 183.542 6 1.366 24 180.370 7 Daytona International Speedway 2.5 31 191.188 8 Dover International Speedway 1 24 157.315 9 Homestead-Miami Speedway 1.5 20 176.904 10 Motor Speedway 2.5 9 182.278 11 1.5 15 174.644 12 1.5 20 188.719 13 .526 12 97.018 14 International Speedway 2 18 189.984 15 New Hampshire Motor Speedway 1.058 12 133.572 16 Phoenix International Speedway 1 10 136.389 17 2.5 9 171.393 18 Richmond International Raceway .750 14 127.077 19 2.66 33 184.640 20 1.5 24 195.397

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.

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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?

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￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿ ￿ ￿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￿ , 9. ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ 5. Graph the scatterplot of your data below. Be sure to label each axis.￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿

Page | 3 TI-83 Symbol Fonts Download font from: http://education.ti.com/educationportal/appsdelivery/download/download_eula.jsp?applicationId=6179&contentPaneId=13&cid=US Shift (Caps) ~ ! @ # $ % ^ & * ( ) _ + ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ Q W E R T Y U I O P { } | ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ A S D F G H J K L : “ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ TI-83 Symbol Fonts Download font from: http://education.ti.com/educationportal/appsdelivery/download/download_eula.jsp?applicationId=6179&contentPaneId=13&cid=US Shift (Caps) Z X C V B N M <~ ! >@ # ? $ % ^ & * ( ) _ + ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿￿ ￿ ￿￿ ￿ ￿ ￿ Q W E R T Y U I O P { } | Normal keys ￿ ￿ ￿ ￿TI-83 ￿Symb￿ol F￿onts ￿ ￿ A S DownloadD font fromF: http://educatG ion.ti.comH /educationporJ tal/appsdelKivery/downlLoad/download_e: ula.jsp?app“ licationId=6179&contentPaneId=13&cid=US ` 1 2 3 4 5 6 7 8 9 0 - = Shift (Caps) ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ~ ! @ # $ % ^ & * ( ) _ + Z X C V B N M < > ? ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿ Q ￿ W￿ E￿ R￿ T￿Y ￿ U ￿ I￿ O￿ P { } | q w e r t y u i o p [ ] N\ormal keys ` 1 2 ￿3 4 ￿ 5 6 7 8￿9 ￿ 0 ￿ - ￿ = ￿ ￿ ￿ A S D F G H J K L : “ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ q w e r t y u i o p [ ] \ Z X C V B N M < > ? a s d f g h j k ￿l ;￿ ￿ ‘ ￿ ￿ ￿ ￿ ￿ ￿ Building SPEED￿ ￿ ￿ ￿ ￿ ￿ ￿What Makes a ￿ ￿Track Fast?￿ a s d f g h j k l ; ‘ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ Normal keys ￿ ￿ ` 6.￿ Does1 ￿ there2 appear￿ 3 ￿to be a4 correlation5 between6 ￿ the7￿ banking8￿ degrees9 and0 the - = z ￿ x speed￿ c of the￿ v track?￿ b Why orn why￿ not?m ￿ If yes,, ￿ is the. ￿ correlation/ ￿ positive￿ or ￿ ￿￿￿ z x c v b n m , . negative?/ ￿ q w e r￿ t y ￿u i￿o ￿ p [ ] \ ￿ ￿ ￿ ￿￿ ￿Ex￿tend￿ed Characters￿ ￿ ￿ ￿ ￿ ￿ (These characters are accesse d by holdinga downs the d“Alt” kefy then tygping theh numbersj on the numerk ic keyl pad.) ; ‘ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ Extended Characters ￿ ￿ ￿ ￿ z ￿x ￿c v￿ b ￿n ￿m , . / (These characters are accessed by holding down the “Alt” key then typing the numbers on the numer ic ￿key pad.) ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿Extend￿ed Chara￿ cters￿ ￿ (These characters 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 −0.1 0.1 to 0.3 coefficient (r-value) for this data? Do you think this Medium −0.5 to −0.3 0.3 to 0.5 is a strong, moderate, or weak correlation? Describe Large −1.0 to −0.5 0.5 to 1.0 the relationship between these two numbers. (you ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ may need to turn￿ “Diagnostics” on)

￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿

￿ ￿ ￿ ￿ ￿ ￿ Page | 4 Building SPEED What Makes a Track Fast?

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.

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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.

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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 + 143.107 8. r = 0.336, there is a medium positive correlation 9. Track length and speed have a stronger correlation at r = 0.730 10. Those tracks have longer straightaways are more opportunities for gaining higher speeds.

Extension: 11. No, it appears to have a curved shape 12. y = –38.369x2+156.292x+33.563 13. R2 = 0.869, so R=.932 much better fit! 14. 181.670 mph (For the inaugural race, qualifying was rained out, so we will have to check this speed next summer!)

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