Determining an Appropriate Parabolic Model
In this procedure, I am trying to determine an appropriate parabolic model that fits the data I collected of whirlybird wing length vs. time, doing so by using first principles. Also to find out which wing length would produce longest flight time.
Method:
Firstly, I made a whirlybird model and timed how long it took to reach the floor from a certain height. This procedure was repeated several times, each time lessening the wing length and keeping the same height. For each wing length, the bird was dropped three times for maximum accuracy. Once this data was collected it was transferred into a graph, and strange points were excluded. After this an appropriate
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As a gets closer to 0 the parabola becomes wider. The result was still too thin.
The next function was:
Y= -0.05(X-8)2+2.33
I changed -0.1 to -0.05. As a gets closer to 0 the parabola becomes wider. The result was still too thin.
The next function was:
Y= -0.03(X-8)2+2.33
I changed -0.05 to -0.03. As a gets closer to 0 the parabola becomes wider. The result was off centre to the left.
The next function was:
Y= -0.03(X-8.5)2+2.33
I changed -8 to -8.5. As b decreases the turning point moves to the right. The result was a little too wide.
The next function was:
Y= -0.035(X-8.5)2+2.33
I changed -0.03 to -0.035. As a gets farther from 0 the parabola becomes thinner. The result was a little too thin.
My final function was:
Y= -0.033(X-8.5)2+2.33
I changed -0.035 to 0.033. As a gets closer to 0 the parabola becomes wider.
Below the final quadratic function, Y= -0.033(X-8.5)2+2.33 is superimposed onto my original data points.
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In the tables below, the results when the different lengths are substituted into the function, can be compared with our initial
Our predicted points for our data are, (13, -88.57) and (-2, -29.84). These points show the
I expect my graph to look like it does on the next page because I
0.96, 0.96, 0.97, 0.98, 1.01, 1.01, 1.02, 1.03, 1.03, 1.03, 1.03, 1.04, 1.04, 1.04, 1.04, 1.05, 1.05, 1.06, 1.07, 1.07, 1.08, 1.09, 1.09, 1.09, 1.09, 1.09, 1.09, 1.10, 1.10, 1.10, 1.10, 1.11, 1.11, 1.11, 1.11, 1.12, 1.16, 1.17, 1.17, 1.18, 1.18, 1.20, 1.21, 1.21, 1.21, 1.23, 1.26, 1.29, 1.31, 1.32, 1.66
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The observation points, days of the week, are marked on the x axis and the frequency of PBA episodes is plotted on the y axis.
This graph shows the result that I expect to get, I expect to see a
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0.000 7 63 106 55 74.7 1.245 9 70 135 90 98.3 1.638 11 85 135 70 96.8 1.613 [ IMAGE ] [ IMAGE ] Conclusion = = = =
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As predicted, the higher the concentration of groundwater pollution, the higher the number of reported mutations and disease cases over the last 10 years was. To add on, the plot points move in an upward direction, which supports my conclusion. In addition, when I conducted my linear regression test on Graph 2, the line moved in an upward direction which reveals a positive correlation. This proves that the higher the groundwater pollution, the higher the number of reported mutations and disease cases will be. Consequently, the correlation coefficient of the graph was 0.99, which is a very strong correlation since it is only 0.01 away from 1. Ultimately, when I created my residual plot graph, the points were closely scattered and close to the X axis. This means that the linear model that I made for the graph was
When looking at the data from the following chart, one can appreciate how many different analyses can be made.