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.
_________ Calculate the sample standard deviation sx.
_________ Calculate the sample Coefficient of Variation.
_________ Determine the class width. Use five classes (bins or intervals).
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 f
Rel Freq p(x)
Sums:
Sketch a histogram of the relative frequency data.
__________________ What is the shape of the distribution?
__________________
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
_________ Is the z-score for
2.06 ms⁻¹ an ordinary or extraordinary value?
_________ Calculate the standard error of the sample mean x
_________ Find tcritical for a confidence level c of 95%
_________ Determine the margin of error E for the sample mean x.
Write out the 95% confidence interval for the population mean μ
p(_____________ < μ < ___________) = 0.95
_________
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 ?
___________
Using the speed data above and a population mean μ =
2.5 ms⁻¹
determine the t-statistic.
___________
Using the speed data above and a population mean μ =
2.5 ms⁻¹
determine the p-value.
___________
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.
___________
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?
count 12
min 2.06
max 2.56
range 0.5000
midrange 2.31
mode 2.42
median 2.4
mean 2.370833
stdev 0.15
cv 0.0645
classes 5
width 0.1000
cul f rf
2.160 2.16 1 0.08
2.260 2.26 2 0.17
2.360 2.36 2 0.17
2.460 2.46 3 0.25
2.560 2.56 4 0.33
12 1
z-scores 2.06 -2.03
z-scores 3.09 4.7
2.68 2
se 0.04
tc 2.2
E 0.1
2.27 ≤ μ ≤ 2.47
2.274 ≤ μ ≤ 2.468
t -2.92 versus μ 2.5
p 0.01 p 0.014
max c 0.99 max c 0.986
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 has changed since last year.
September 2008
October 2009
2.54
2.35
2.54
2.56
2.65
2.42
2.50
2.42
2.55
2.38
2.34
2.28
2.87
2.25
2.65
2.56
2.53
2.06
3.09
2.19
2.49
2.49
Partial solution set
stat
value
stat
value
n
10
n
12
mean
2.63
mean
2.37
stdev
0.21
stdev
0.15
ttest
0.01
max c
0.99
_________ Calculate the sample mean
speed for September 2008 using the above data.
_________ Calculate the sample mean
speed for October 2009 using the above data.
_________ Are the sample means for the two samples mathematically different?
__________________
What is the p-value? Use the TTEST function with two tails
to determine the p-value for this two sample data.
__________________ Is the difference in the means statistically significant
at a risk of a type I error alpha α = 0.05?
__________________ Would we fail to reject| or |reject a null hypothesis of no difference
in the sample means?
__________________ What is the maximum level of confidence we can have that the
difference is statistically significant?
__________________
Has my speed changed, slowed down, by a statistically significant difference over the past year?
Part III: Linear Regression (best fit or least squares line)
y = -0.00696x + 2.416
Data table
Location
Distance (km)
Speed (ms⁻¹)
PICS track
1
2.35
SEI
2
2.56
Near 4TY
3
2.42
3 Star
4
2.42
Nett bridge
5
2.38
Beyond Koahn
6
2.28
Nett bridge
7
2.25
Past 3 Star
8
2.56
Causeway
9
2.06
Mesenieng
10
2.19
Capitol hill
11
2.49
0
12
2.49
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.
_________ Calculate the slope of the linear regression (best fit line).
_________ Calculate the y-intercept of the linear regression (best fit line).
_________ Is the relation between distance and speed positive, negative, or neutral?
_________ Calculate the linear correlation coefficient r for the data.
______________ Is the correlation none, weak/low, moderate, strong/high, or perfect?
______________ Determine the coefficient of determination.
______________ What percent in the variation in
the distance
"explains" the variation in
the speed?
_________ Use the slope and intercept to predict
the speed for a 21 kilometer half-marathon distance.
_________ Use the slope and intercept to determine
the distance I would be predicted to run at 2.4 kph.
_________ If I run zero kilometers, what speed does the regression predict?
_________
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?
_________
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?
count 12
slope: -0.00696
intercept: 2.42
correl: -0.16
coef det 0.03
y given x 2.27 m/s
x given y 2.31 km
Is fifty slower (part II) and less consistent (part I and III)?
Yes fifty is slower (II), yes fifty is less consistent(I), but no, fifty is not slower with distance (III).
