Lab0M_Report_v4 (1) 2

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University of Texas *

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105M

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Statistics

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Feb 20, 2024

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Valerie Mendez (vam2536) Lab 0 Part 1: Is the Marble a Sphere? Method: We are testing whether the marbles provided are spheres. While they seem like spheres from their appearances, we still must confirm through our data. We used a tape measure to quickly measure the diameters of three marbles: a large blue and yellow marble, and two other colored marbles (blue and white) of the same size. In order to account for overlaps in uncertainty, three diameters per marble were measured in inches and compared using the formula that states a marble with equal diameters is spherical (D1=D2=D3). The results were as follows: For more accurate measurements, we measured the diameter of the same marbles again using calipers and a micrometer. We recorded three trials for three distinct diameters for every marble, for a total of nine trials per marble. The conclusion on sphericity would subsequently be reached by comparing the averages of each diameter. By using this procedure, the random uncertainty for each measured diameter as well as the overall result would be reduced. As a result, the data's stated source of uncertainty is systematic. Data : The data was gathered by measuring three different diameters from the marble with three different measuring tools. Table 1: Marble Diameters using Tape Measure Blue and Yellow Marble Blue Marble White Marble Trial Diameter (In) Uncertainty (In) Diameter (In) Uncertainty (In) Diameter (In) Uncertainty (In) 1 0.6692 0.0005 0.5118 0.0005 0.5117 0.0005 2 0.6682 0.0005 0.5511 0.0005 0.5120 0.0005 3 0.6690 0.0005 0.5314 0.0005 0.4724 0.0005 Table 2: Marble Diameters using a Dial Caliper Blue and Yellow Marble Blue Marble White Marble Trial Diameter (In) Uncertainty (In) Diameter (In) Uncertainty (In) Diameter (In) Uncertainty (In) 1 0.6342 0.0005 0.6724 0.0005 0.5297 0.0005 2 0.6331 0.0005 0.6662 0.0005 0.5242 0.0005 3 0.6326 0.0005 0.6708 0.0005 0.5326 0.0005 Table 3: Marble Diameters using Screw Gauge Micrometer Blue and Yellow Marble Blue Marble White Marble Trial Diameter (In) Uncertainty (In) Diameter (In) Uncertainty (In) Diameter (In) Uncertainty (In) 1 0.6312 0.0005 0.5874 0.0005 0.5252 0.0005 2 0.6322 0.0005 0.5908 0.0005 0.5268 0.0005 3 0.6289 0.0005 0.5982 0.0005 0.5253 0.0005 Conclusion : The hypothesis that all the marbles are spheres is not supported by our data. These results were contradicted by the primary experiment, which was verified by decreased uncertainty brought about by more

Valerie Mendez (vam2536) Lab 0 trials and increased accuracy. Blue and Yellow marble had average diameters of 0.67 +/- 0.0005 in, 0.64 +/- 0.0005 in, and 0.63 +/- 0.0005 in. As the second and third diameters do not overlap, they cannot be considered equal, so the blue and yellow marble cannot be spherical by the formula D1=D2= D3. Small blue marble had average diameters of 0.53 +/- 0.0005 in, 0.67+/- 0.0005 in, and 0.59 +/- 0.0005 in so the blue marble is not spherical. The white marble had average diameters of 0.49 +/- 0.0005 in, 0.53 +/- 0.0005 in, and 0.52 +/- 0.0005 in, so the white marble is not spherical. A few possibilities for the next iteration of the experiment would be to use the micrometer for more accurate precise measurement, measure more of the diameters from the marbles for more data and use new unchipped marbles to get more accurate data. Should we sue the manufacturer? I don’t think so, because marble is still in a general area when compared to a sphere, so I don’t think it's far off even if its diameters do not match. Part 2: Does Color Influence Rolling Time? Method : The goal for this part is to determine the dependence of a marble's rolling time down a ramp on its color and size. We predict color has no effect on rolling time, while the size difference does. The two same-sized marbles, as previously confirmed, are not precisely spherical; their deviance from sphericity will likely affect their respective rolling times. Using the same marbles from the previous experiment, we fixed a wooden ramp at an incline and placed an object at the end as a stopper. Then, we recorded the time it took for one of the small marbles to roll down the ramp and repeated for a total of 5 trials each. We then repeated the procedure for the second, differently colored(white) marble, then again for the extra marble (blue). The uncertainty values cited in the data are the standard deviations of the means of the trials. We predict that there won't be a difference in the rolling times between the two marbles because none of these factors are dependent on color. We're going to use the stopwatch to measure time. Every time, we will merely change the marble's color; the ramp's length and angle will remain constant. We will begin by timing how long it takes a marble to roll down a ramp five times, from which we will calculate the mean and standard deviation. A few of the influences on the marble as it rolls down the ramp include gravitational force, friction with the ramp and/or the air, angle of the incline, and wobble of the incline. Data : The seconds each marble produced from rolling down the ramp. Table 4: Rolling times for Marbles Blue and Yellow White Marble Blue Marble Trial Time (s) Uncertainty (s) Trial Time (s) Uncertainty (s) Trial Time (s) Uncertainty (s) 1 2.44 0.01 1 2.01 0.01s 1 1.85 0.01 2 2.25 0.01 2 1.91 2 1.92 0.01 3 2.00 0.01 3 2.21 3 2.11 0.01

Valerie Mendez (vam2536) Lab 0 4 1.90 0.01 4 2.24 4 2.03 0.01 5 2.01 0.01 5 1.84 5 2.20 0.01 Mean 2.12 Mean 2.0 Mean 2.0 STD 0.22 STD 0.17 STD 0.11 Conclusion : The difference between the two marble’s mean times to roll down the ramp is . 12s. Each has an uncertainty of .01s so the two results overlap. The blue and yellow marble took an average of 2.12 +/- 0.01s to roll down the ramp, while the white marble took an average of 2.0+/- 0.01 s. Thus, color does not matter. The marble's size does, however, as the extra marble had a rolling time of 2.0+/-0.01 s, which is significantly different. This result confirmed our hypothesis as the extra marble and white marble were of the same size. If we were to conduct another iteration of the experiment, we would want to reduce the uncertainty further by performing more trials per marble. Marking the exact start location would also reduce random uncertainty. We could then compare, as stated before, the dependence of rolling time on sphericity.

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