From MariachiWiki

While statistical experiments tend to be tedious, we came up with an innovative way to transform statistics and random processes into a fun learning experiment. We designed an experiment that explores statistics with the use of ordinary microwave popcorn. The sound of the popping kernels would provide the necessary data to complete the lab exercise.

Contents 1 Popcorn Physics 2 Materials 3 The Experiment 4 Analysis 5 Interpretation of Results 6 Discussions

Popcorn Physics

The science behind microwave popcorn can be described in the following manner. Each kernel will explode when it is heated to high enough temperature to evaporate the water inside the kernel. This creates an immense amount of internal pressure which causes the kernel to burst. Therefore, old kernels would not burst.

But why "microwave" popcorn? The old fashioned way to make popcorn is to put kernels in a pan with butter, oil or fat. The heated oil distributes the heat more efficiently and evenly. Thus, it serves as a better catalyst for the "popping effect" of the kernels. Similarly, microwave popcorn combines the kernels and the carrier source of fat in one package. This makes the reaction much more effective.

Materials

• Microwave oven
• Microwave popcorn bag
• Microphone
• Computer with sound card or laptop
• Audacity

The Experiment

One goal of this experiment (as an example) is to determine the distribution of the number of pops as function of time. There are some alternative experiment that can be done to demonstrate the same ideas. These will be expanded upon later. To perform the measurement we place a microphone next to a microwave oven and record the sound while kernels are popping. Then the sound is analyzed and the time of each pop is tabulated using Microsoft Excel. In this case we were able to use an open source software Audacity to both record and analyze the data. Audacity is available for all computer platforms.

Once the bag of microwave popcorn is placed inside the microwave it is important to start the recording before the microwave. Th reason for this, is that the information for time zero is very important. The position of the microphone is also crucial. A proper orientation will enhance the ratio of signal versus noise. Microwave ovens, especially older models, have high level of fan noise. We found that a microphone positioned directly in front of the window will produce the best results. Yet, be ware that some experimentation may be required to find the exact spot. If this is not done properly, the poor signal to noise ratio may prove problematic in this experiment. Different sampling rates for the sound recording can be used depending on your computer. The lowest sampling rate of 8000 ksample per second is found to be too low, whereas a 44000 ks/s was found to be too high. Therefore either 14ks/s or 22ks/s should be used. It is more than adequate to use 16 bits digitization. The overall recording time for this paper was 3:10 min. However this may depend on the type of microwave and power.

Analysis

Audacity is also used for data analysis. The software allows for the manipulation of sound files (.wav) for further noise reduction. This increases the accuracy of the counts. The first step in the data analysis is to investigate the different frequencies that were recorded. With the help of previously recorded microwave noise, frequencies can be determined. A notch filter is used to reject those noise frequencies, leaving behind only the frequencies of the popping sound. There are two ways to count pops. The first way is to count directly looking at the recorded waves. Otherwise a spectrogram can be used to plot sound frequency against time. We found the latter more accurate.

To count accurately one needs to zoom in an appropriate time window to distinguish close pops. At times up to 4 kernels pop nearly simultaneously making counting difficult. We found that in these cases it is really more accurate to listen and count the number of pops. These instructions are general remarks and each run will require slightly different analysis technique to reach accurate counting. We have also counted the number of kernels that did not pop and they amount to 38 (after eating all the popcorn). This is an interesting number by itself because the number of pops counted adds to 266. This means that 12.5% of kernels did not pop.

We graph the number of pops for each time bin of 10 seconds each, taking the square root of number of counts as an estimate for the uncertainty for that bin. We use Microsoft Excel for the analysis for being software that is readily available. To extract the relevant parameters we made an assumption that the distribution is Gaussian. A full χ² fit is then made resulting in an overall reduced χ² of 0.6, indicating that the assumption of a Gaussian distribution is somewhat appropriate but perhaps other distribution might be better. It is also true that the overall number of events is small (~300). The mean value is (69±2) with a sigma of (31±3).

Interpretation of Results

Data show that no kernel pops before 1:36 min from start of the experiment. This time corresponds to the minimum interval required to heat a kernel to pop. After that the number of pops gradually increases reaching a maximum 70 s after activity start and declines. By the time the number of pops decrease the experiment ended. This observation suggests the following cycles for the process: (1) heating cycle, where all kernels are heated (2) popping cycle, when kernels pop but with continuing heating and (3) the final cycle where external action is turned off and only residues are left. Looking at the distribution of pops, one can speculate that either each kernel requires different temperature to pop, or they require the exact same temperature and they reach the critical temperature at different times. To answer this question a separate controlled experiments with popcorn only should be made.

Based on the number of kernels that popped we can estimate the number of kernels that didn’t pop as the area under the curve after the experiment is finished. This estimative give us (7±3) kernels compared to 38 that we counted. We can concluded that either there are kernels that would not pop at all, or that whatever mechanism used to heat the kernels to pop is not fully efficient. This depends on the internal arrangement in each popcorn bag. This effect could also be instrumental. The microwave oven used didn’t have a turntable, which could make a large difference because it homogenizes sample heating. Inside a microwave oven standing waves exist and by rotating the object heat is evenly distributed and you don’t rely as much in temperature convection. We performed a separate experiment where the bag is preheated for 30 s followed by a no power cycle of 2 min, and then full power for 3 min. In this case most of the kernels pop. This favors the explanation that the reason why there were so many kernels left is because heat is not evenly distributed.

Putting all the results together we come with the following picture of what is inside the microwave popcorn bag. Inside the bag there are to kernels and some sort of media that help homogenize heat. We will call them a carrier. By pre-heating the bag the carrier heats up and transfer the heat to the popcorn to pop. This could be some sort of organic fat that melts in the microwave oven. The observation that different heat cycles reduce the number of un-popped kernels supports this idea. Different kernels must have different temperatures to pop under this scenario. Fat when liquid conducts heat very efficiently, therefore we can assume that all kernels are being heated to similar temperatures. The results can lead to other experiments to corroborate conclusions made here. Experiments, especially those that refer to the subatomic universe also require a lot of information before hand. We are not allowed to see what are inside atoms but we infer the internal structure by indirect measurements.

Discussions

A very simple experiment is described on how to perform a statistics experiment. The experiment uses open source software, Audacity, a computer and microwave oven. Most items are easily found. The data acquisition part of the experiment requires short time but careful positioning of microphone and knowledge of background required. Analysis can be performed manually and in this example we plotted the distribution of number of pops as function of time. Other interesting studies can be done with the same setup. For example the interval between pops could be examined, or the probability of having multiple pops in a given time internal, and so on. A lot of features in this experiment are the same as used in may physics experiments, especially those that require counting, e.g. measurement of cross section in nuclear physics.

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