HYPOTHESIS
-
Alternative Hypothesis 1: Power during a 100 m sprint does change over the course of a day
-
Null Hypothesis 1: Power during a 100 m sprint doesn't change over the course of a day
-
Alternative Hypothesis 2: Given the same time of day, individuals will exert a different amount of power
-
Null Hypothesis 2: Given the same time of day, individuals will not exert a different amount of power
DATA ANALYSIS
The purpose of the data analysis is to have a more comprehensive view of our data. Our fist analysis was done using ANOVA and t-tests to show if there were any significantly different variables with respect to the time of day. We performed a correlation and regression analysis in order to see if we could actually imply causation due to any correlation between variables. We also performed a sensitivity analysis to see how our variables in the project equations would tend to be affected by changing values. This would help us understand the best way to collect the most accurate data in the future.
STATISTICAL ANALYSIS
By Marissa Kuhns
Using the percent exertion obtained in the experiment for each individual, as shown in Figure 6, graphs (Figures 7-10 below) were created to show the spread of the data for each individual compared to each other at a specific time.
The ANOVA tests performed were two-factor with replication. This is because we had to compare more than one variable to a single factor, for example we were comparing an individual to the time of day and their power output. Likewise this was done for percent exertion as well.
CORRELATION and REGRESSION ANALYSIS
By Ashley Hah
Correlations and regression analysis was carried out to study the correlation of variables of interest and formulate a trend line for those specific variables. The desired outcome of conducting a regression and correlation analysis is to be able to draw a conclusion to proof whether the performance of an individual is influenced by the time of day in which workout is done. As mentioned in our project overview, the power output and % exertion was determined by the following equation:
The equation predicts higher power output for subjects with shorter time of sprint for a given distance. However, it is also important to note that power output is dependent on the mass of the subject, which makes it hard to compare the power output between individual. This observation can be proven graphically when the power output of all three subjects was plot against time of day (Figure 1) and when %Exertion of all three subjects was plot against time of day (Figure 2). There was no trend line that can be concluded from the graph. To prove the point, regression analysis was done on the overall data of all three subjects and the P-value shows no significant correlation between power, time of day and heart rate.
Figure 15 Power vs. time of day plot for all three subjects
Figure 16 % exertion vs. time of day plot for all three subjects
With that said, the correlation and regression analysis was done for each subject. The correlation coefficient for each variable was calculated using the data collected for each subject and the results are as shown below (Figure 17). Based on the values of r obtained, we found a few strong positive and negative correlations between variables. However, for the purpose of this experiment, we will only be focusing on the relationship between power output, % exertion and time of day.
Figure 17 Correlation coefficient for each variable was calculated for each subject. Cells highlighted in green shows variables with strong negative correlation and cells highlighted in orange shows variables with strong positive correlation.
Figure 18 P-values for each variable was calculated to determine the significance of the correlation. Cells highlighted in green shows variables with significant correlation. Cells in red are variables that are actually considered in for the purpose of this study
CORRELATION ANALYSIS
REGRESSION ANALYSIS
Ashley
Matt
Marissa
Results of Correlation Analysis
Based on the correlation coefficient found through this analysis, we found that the correlation of each variable differs between each subject. This is an interesting finding because we predicted that there would be some sort of pattern across our data. Follow up t-statistics was done to obtain the P-values for each correlation coefficient in order to identify which correlations are significant. Results obtained from the P-values showed a few variables with significant correlation. However, as mentioned above, regression analysis will only be carried out on variables that will help answer the project question, these variables are power, heart rate and time of day of workout
Ashley
Regression analysis was done separately for each individual and P-value for power with respect to heart rate and time of day was obtained. A trend line was developed for Power vs. Tday plot and Power vs. Heart Rate plot for each subject. It is important to note that since every test subject is different, the results obtained from one subject cannot be used to generalize the result of the whole experiment.
Regression Analysis Result for aSHLEY
The graphs above shows the relationship between power vs. time of day and % exertion vs. time of day. For the plot of power vs. time of day, a polynomial trend line was developed with a R-square value of 0.8207. As for % exertion vs. time of day, the R-square value is much smaller since the data is much more spread out especially for % exertion at 12 pm. There is not really a trend observed based on the data obtained, however, it can be concluded that the level of performance for this subject is optimum at 8 am, considering the high power output and high % exertion. The P-value for these variables shows significant correlation as shown in the table below
The regression analysis also made it possible to develop an equation that captures the relationship between power, time of day and % exertion for subject to calculate the her estimated power output for a given workout. The equation is as follows:
Matt
Regression Analysis Result for mATT
A R-square value of 0.956 was obtained for the plot of power vs time of day, showing a good fit. On the other hand, a R-square value of 0.815 was obtained for the plot of % exertion vs time of day. The high R-square value shows that there is some sort of trend in the data set for this subject. Based on the graphs, we can say that this subject's workout performance increases as time of day increases as the power output and % exertion shows the tendency to increase with time. It can also be concluded that the level of performance of this subject is optimum at 8 pm considering his power output and % exertion to be highest at that time of the day. Similar to Ashley's results, the P-value for % exertion, power and time of day showed a significant correlation as shown below
An equation that relates power, % exertion and time of day was developed as shown below to enable this subject to estimate his power output for a given workout
mARISSA
Regression Analysis Result for mARISSA
Marissa's data points are more widespread as compared to the other two subjects. The spread of the data is the reason why the R-square value for her trend line to be lower as compared to the other subjects. With that said, there is not specific pattern to the data points. However, we can conclude that the time of day in which her performance is at optimum level is at 3 pm considering her power output to be the highest at that time. As for the P-value for power, % exertion and time of day, the number is much higher as compared to the other two subjects implying that the correlation are not significant. It makes sense considering her power outputs are in a wider range.
From regression analysis, an equation that relates power, % exertion and time of day was developed based on the data obtained from the experiment. The equation is as follows:
Figure 19 Trend line for power vs. time of day and % exertion vs. time of day plot
Figure 20 Trend line for power vs. time of day and % exertion vs. time of day plot
Figure 21 Trend line for power vs. time of day and % exertion vs. time of day plot
ANOVA and t-tests
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
When looking at the ANOVA results for power (Figure 11), all the p-values are below alpha, which was 0.025. With this information it allowed us to conclude that our means were statistically different at different times of the day. Our F value was also greater than our F critical value, which allowed us to conclude that there is a significant difference between means of at least 2 samples. To follow up with this, t-tests were performed to compare each mean at each time of the day for each individual.
When looking at the ANOVA results for percent exertion (Figure 12), all the p-values are below alpha, which was 0.025. With this information it allowed us to conclude that out means were significantly different at different times of the day. Our F value was also greater than our F critical value, which allowed us to conclude that there is a significant difference between means of at least 2 samples. To follow up with this, t-tests were performed to compare each mean at each time of the day for each individual.
Figure 13
Figure 14
T-tests were done as the next step in the ANOVA process, where a follow-up t-test between each pair of samples was done. You can see this in Figures 13 and 14, where it compares each time of the day to another time of the day and compared the means in the power output from Figure 1 and the means of the percent exertion from Figure 6. If the p-value was less than alpha then you can conclude that that there is a significant difference in the means of those two times.
Here is a link to the full statistical analysis Excel Spreadsheet:
Using the power values obtained in the experiment for each individual, as shown in Figure 1, graphs (Figures 2-5) were created to show the spread of the data for each individual compared to each other at a specific time. These groaphs also provide continuity with the Project Overview Tab above and show the actual results of the experiment.
Results from ANOVA And T-tests
Based on the results from Figures 13 and 14 we can now see if there is an optimum time of the day for an individual to work out where their performance will be at a peak. For Ashley 8:00 AM would be the suggested time because when comparing 8:00 AM to other times of the day in Figures 13 and 14 the means are significantly different. For Marissa, there was no trend in her values, and therefore we cannot provide a recommended workout time for her. Finally for Matt, a workout time of 8:00 PM would be the most optimum time due to the significantly different means when dealing with the 8:00 PM time slot.
The purpose of this statistical analysis is to compare different sets of our data to suggest an optimum time of day for an individual to workout. In this section, ANOVA and t-tests were used in order to compare more than two sets of data. ANOVA shows if there is a significant difference between means of at least 2 samples. Then it is followed up with t-tests in order to see if there is a significant difference between means of samples.
Figures 7-10 show the average percent exertion for each individual at a given time of the day. Error bars represent the standard deviation.The bars highlighted in yellow show that at that time for that individual, the means for percent exertion were significantly different.
Figures 2-5 show the average power output for each individual at a given time of the day where the error bars represent the standard deviation. The bars highlighted in yellow show that at that time for that individual, the means for power output were significantly different.
Results from Power and percent exertion graphs
Both power and percent exertion are the ways chosen to measure peak performance as well as optimum time of day for an individual to workout. All of these graphs help to provide a visual of each individual compared to themselves as well as the other two participants.
STATISTICAL ANALYSIS
By Marissa Kuhns
By: Matthew Tomasetti
SENSITIVITY ANALYSIS
The purpose of this sensitivity analysis is to determine how much certain parameters affect the power output during a 100 m sprint, and therefore, the overall performance of a workout. We took the average values for each test subject's respective mass, velocity, and time taken to complete the sprint and used them in the governing power equation:
Next, we calculated the power exerted during a workout at respective low, medium, and high values for mass, velocity and time of sprint. Then, we averaged the actual power output values obtained in our data collection for low, medium, and high times of the day ( 8 am, 12 pm, and 8 pm). These were plotted along with the other values for the three variables mentioned above. Looking at the visual results of the line graph and bar graph of the % baseline enables us to get an idea of how "sensitive" our multi variable power equation is when we change the values of each of the terms.
Sensitivity Analysis Data
The average values for each of the three parameters were selected as low, medium, and high. (represented in Table 1.1 below) The corresponding power was calculated using the medium values
Next, we varied each of the variables (mass, velocity, and time) individually, while holding the other variables at their medium baselines. The power was calculated for each of these, and the overall % change in power, as compared to the medium values (Shown in Tables 1, 2, and 3 below)
Table 1.1
Next, the graphical results of the sensitivity analysis were obtained in order to have a visual representation of how changing the variables in the power equation affects the overall power output of a workout.
Figure 1: Sensitivity of Power output by numerical value
Figure 2: Sensitivity of Power output in percentages
Table 1
Table 2
Table 3
Sensitivity Analysis Results
From the sensitivity analysis shown, one can see that to obtain the goal of producing the highest power output during the day, it is necessary to maximize both the velocity, and mass of the person, while minimizing the time taken to complete the sprint. Having the highest mass possible will show a numerically higher power output, however, this cannot always vary drastically over the course of a day. There is also a correlation that the power output increases over the course of day and hits a "peak in the middle" and then decreases slightly at night. Also, as common sense dictates, running the distance at the highest speed (increase velocity) and the shortest amount of time, has a big influence on optimizing performance.
Complete spreadsheet of Correlation and Regression analysis
Figure 1 shows the power output in Watts for each individual at each of the 4 times of day along with the average and standard deviation.
Figure 6 shows the percent exertion for each individual at each of the 4 times of day along with the average and standard deviation.
