Part I: Activity

Here, ⍺ is the mode of the distribution. This is the value which appears most often in the set. To complete the activity, please follow these steps.

1. Imagine now that you throw 20 darts at the board. Generate 20 pairs of random values for the x and y values of the darts, using the random number generator below. Create values with three significant figures (in this case, two decimal points; for instance: 8.39.) You may also use Excel or Numbers to generate random numbers; for instance, the Excel function RAND() produces a random number between 0 and 1.

2. Input these values into Excel or Numbers. Excel is suggested, since I give directions below only in Excel.

3. Compute r for each dart. 

We will assign the following point values to the throws: 

0< r < 2: 1.95 points

2< r < 4: 1.66 points

4< r < 6: 1.21 points

6< r < 8: 0.76 points

8< r < 10: 0.40points

10< r < 12: 0.18 points

12< r < 14: 0.07 points

r>14: 0 points

Now create a separate column entitled "Point Values"; to obtain these, multiply each r value by its point value.  Produce a histogram of your data which separates the point values into the following ranges (these are often called "bins"): 0 to 1, 1 to 2, 2 to 3, .... To accomplish this in Excel, first create the bins by creating a column consisting of integers from 1 to the maximum point value. Then, select the "Data" tab and then click "Data Analysis". Select "Histogram". The "Input Range" should consist of the point values. The "Bin Range" should be the aforementioned column of integers. Click "Output Range" and then select a blank cell. A table will be created from this cell. Finally, check "Chart Output". This should produce a table of the frequencies of r values, and an associated histogram.

Now answer the following questions regarding your data set. Please type your answers in a Word (or equivalent) document, which you will submit below. This is an assignment, and not a presentation. 

1. What is ⍺? If there is a not a single mode, average the modes to produce a single value.

2. Given that we are studying a distribution of radii, does this value of ⍺ make intuitive sense?

3. Perform a literature search to find a plot of the Rayleigh distribution, and cite your sources. (Alternatively, you could make a plot of the probability function given above.) Does your histogram look like an exact Rayleigh distribution? If not, how could this experiment be tweaked to produce a better facsimile? Does this say anything about the usefulness of statistical distributions with different sample sizes?

Part II: Application to Nutrient Response

Please open and read this paper and answer the following questions based upon it.

1. In your own words, what do you think is the main hypothesis of the paper?

2. Describe how the authors attempted to falsify the hypothesis.

3. In Figure 1, there are six figures. The data is plotted with error bars, which the caption indicates are standard errors. Look up the definitions of  standard error and standard deviation. In a random, unbiased data set, what the probability that a randomly selected point will fall with one standard deviation of the mean? What is the probability that such a point would fall outside of one standard deviation from the mean?

4. Do you think the study convincingly demonstrates that the hypothesis is true?