Is fifty slower and less consistent? • Name:

LocationSpeed (ms⁻¹)
PICS track2.35
SEI2.56
Near 4TY2.42
3 Star2.42
Nett bridge2.38
Beyond Koahn2.28
Nett bridge2.25
Past 3 Star2.56
Causeway2.06
Mesenieng2.19
Capitol hill2.49
Dolihner2.49

running the road

Part I: Basic Statistics

In class I often extol the virtues of exercise. Yet if I do not also partake of exercise, then I am like a doctor who will not take his or her own medicine. I have been running for over thirty years. Of late I am not running as much as I did in years past. At fifty my speed may be slowing down, or becoming less consistent. If my speed is less consistent, then that would also be a sign of a lack of sufficient training. On 30 October I went out for a long run of 12 kilometers, recording my split times for each kilometer. The table reports my calculated speed for each split.

Data sheet

  1. _________ What level of measurement is the data?
  2. _________ Determine the sample size n.
  3. _________ Determine the minimum.
  4. _________ Determine the maximum.
  5. _________ Calculate the range.
  6. _________ Calculate the midrange.
  7. _________ Determine the mode.
  8. _________ Determine the median.
  9. _________ Calculate the sample mean x.
  10. _________ Calculate the sample standard deviation sx.
  11. _________ Calculate the sample Coefficient of Variation.
  12. _________ Determine the class width. Use five classes (bins or intervals).
  13. Fill in the following table with the class upper limits in the first column, the frequencies in the second column, and the relative frequencies in the third column
Classes (x)Frequency fRel Freq p(x)
Sums:
  1. Sketch a histogram of the relative frequency data.
  2. __________________ What is the shape of the distribution?
  3. __________________ Out on the causeway I dropped to a speed of 2.06 ms⁻¹ Use the sample mean x and sample standard deviation sx above to calculate the z-score for 2.06 ms⁻¹
  4. _________ Is the z-score for 2.06 ms⁻¹ an ordinary or extraordinary value?
  5. _________ Calculate the standard error of the sample mean x
  6. _________ Find tcritical for a confidence level c of 95%
  7. _________ Determine the margin of error E for the sample mean x.
  8. Write out the 95% confidence interval for the population mean μ
    p(_____________ < μ < ___________) = 0.95
  9. _________ In physical science class I have quoted my speed as 2.5 ms⁻¹ . Based on the confidence interval above, is the mean speed on this 12 kilometer long run statistically significantly different than the μ = 2.5 ms⁻¹ ?
  10. ___________ Using the speed data above and a population mean μ = 2.5 ms⁻¹ determine the t-statistic.
  11. ___________ Using the speed data above and a population mean μ = 2.5 ms⁻¹ determine the p-value.
  12. ___________ Using the speed data above and a population mean μ = 2.5 ms⁻¹ determine the maximum confidence c interval for which the difference is statistically significant.
  13. ___________ The speed for a well trained recreational runner should not vary by more than 5%. A variation greater than 5% is inconsistent and a sign that the runner should get back to the basics of training. Use the coefficient of variation as a measure of my consistency. Am I, at fifty, a less consistent runner in terms of speed?

running the road

Part II: Hypothesis Testing using the t-test

In part two you will run a two sample hypothesis test on whether there is a statistically significant difference between two set of data using a risk of a type I error alpha α = 0.05 Over a year ago, on 03 September 2008 I performed a similar run on a comparable route measuring my split times per kilometer. Use a two-tailed t-test for two samples to determine whether my speed in meters per second has changed since last year.

Speed September 2008 (m/s)Speed October 2009 (m/s)
2.542.35
2.542.56
2.652.42
2.502.42
2.552.38
2.342.28
2.872.25
2.652.56
2.532.06
3.092.19
2.49
2.49
  1. _________ Calculate the sample mean speed for September 2008 using the above data.
  2. _________ Calculate the sample mean speed for October 2009 using the above data.
  3. _________ Are the sample means for the two samples mathematically different?
  4. __________________ What is the p-value? Use the TTEST function with two tails to determine the p-value for this two sample data.
  5. __________________ Is the difference in the means statistically significant at a risk of a type I error alpha α = 0.05?
  6. __________________ Would we fail to reject | or | reject a null hypothesis of no difference in the sample means?
  7. __________________ What is the maximum level of confidence we can have that the difference is statistically significant?
  8. __________________ Has my speed changed, slowed down, by a statistically significant difference over the past year?

running the road

Part III: Linear Regression (best fit or least squares line)

Distance versus speed background rectangle major grid lines axes x-axis and y-axis linear regression line data points as circles text layers Distance versus speed October 2009 Distance (km) Speed (ms⁻¹) y-axis labels 2.06 2.11 2.16 2.21 2.26 2.31 2.36 2.41 2.46 2.51 2.56 x-axis labels 1 2 3 4 5 7 8 9 10 11 12

Data table

LocationDistance (km)Speed (ms⁻¹)
PICS track12.35
SEI22.56
Near 4TY32.42
3 Star42.42
Nett bridge52.38
Beyond Koahn62.28
Nett bridge72.25
Past 3 Star82.56
Causeway92.06
Mesenieng102.19
Capitol hill112.49
Dolihner122.49

Runners tend to run at the fastest speed for which they can still finish the distance. Thirty years of running have trained me to know how fast I can go for a given distance. To go farther I have to run a little slower. This last section of the final explores this relationship between the length of a run in kilometers and the speed at which I run.

  1. _________ Calculate the slope of the linear regression (best fit line).
  2. _________ Calculate the y-intercept of the linear regression (best fit line).
  3. _________ Is the relation between distance and speed positive, negative, or neutral?
  4. _________ Calculate the linear correlation coefficient r for the data.
  5. ______________ Is the correlation none, weak/low, moderate, strong/high, or perfect?
  6. ______________ Determine the coefficient of determination.
  7. ______________ What percent in the variation in the distance "explains" the variation in the speed?
  8. _________ Use the slope and intercept to predict the speed for a 21 kilometer half-marathon distance.
  9. _________ Use the slope and intercept to determine the distance I would be predicted to run at 2.4 kph.
  10. _________ If I run zero kilometers, what speed does the regression predict?
  11. _________ One measure of running fitness is the ability to hold a constant speed for the duration of a run. Based on the strength of the correlation, is my speed changing as the distance increases?
  12. _________ At fifty, does the strength of the correlation show that at fifty I am statistically slowing down with increasing distance? | or | holding a statistically constant speed with increasing distance?
  13. _________ Is fifty slower (part II) and less consistent (part I and III)?

running the road