Homework 2, Testing the Die Class.

Short Description:

Goals

• Worked with vectors.

Formal Description

To test the die class, we will use the Chi-squared test. I have been assured by Dr. Hoggard that this is the proper test to use.

I have found this page to be quite helpful in understanding the test. It includes a very good example. (You will be asked to use this example as a test case later).

In our case, where there are equally likely outcomes, the test is performed by sampling the different categories of the population and seeing how far this sample varies from the expected value the categories. The formula to compute this is:

$$ \chi^2 = \Sigma_{i=1}^{k}\frac{(x_i-m_i)^2}{m_i}$$ Where

• $x_i$ are the results of sampling the population in category i.
• $m_i$ is the expected value for category i.

Since you are testing a uniform distribution $m_i$ is just trials/sides.

Since you are testing a $n$ sided die, find $x_i$ by rolling the die many times and counting how many times i occurs.

You might want to adjust the index i to better work with 0 indexed vectors.

Once $\chi^2$ is compute, you examine the Chi-Square Probabilities Table, using the degrees of freedom (IE categories -1 ) and the desired significance level to determine if your if you reject or fail to reject the null hypothesis that the sides of the die appear to be uniformly distributed.

Work your way through the example on the first link for proper procedure.

You are to produce a tool which will allow a user to conduct a $\chi^2$ test on a vector of results. Assume that your user will pass you a vector of integers, the results of a sampling and that the user expects these results to be uniform.

In specific, you should support

• Performing a $\chi^2$ test on a vector of integers. The vector will have at least 2 entries and at most 31. The user will supply the vector of sample frequencies and an $\alpha$ value and the test will return true or false based on the relationship of the table, $\alpha$ and the calculated value of $\chi^2$.
• Calculate $\chi^2$ for a given vector.
• Look up a value in the $\chi^2$ table for a given $\alpha$ and degrees of freedom (df, 1 ≤ df ≤ 30).
• Get a list of valid values for $\alpha$. This will be a vector of floats, containing the possible values your code supports.
• Provide the ability to validate a given value for $\alpha$, IE report true if you can provide a $\chi^2$ value for that $\alpha$.

To support your system, and to show that it works and test its functionality, please provide a test driver that does the following:

• Conducts the analysis case from the tutorial link above. You should produce results similar to the following:

  Testing the example data
          Stat = 52.750
          X^2(4,0.050) = 9.488
          9.488 < 52.750
          The result is reject.
       

• Conducts an analysis of testing the Die class for dice with sides 2, 4, 6, 8, 10, 12, and 20. Run at least 1,000,000 trials for each test and test at various levels. Example output should be similar to

  Testing a d6 for 1000000 trials.
          Stat = 2.030
          X_(5,0.010) = 15.086
          2.030 < 15.086
          The result is do not reject.
       

Useful Files

• Makefile.
• Die.h
• Die.cpp.
• chiTable.dat, the chi table in csv form, taken from Professor James Jones' page.

Additional notes

• Your code should be in the form of a system which includes a header file and an implementation file.
• All vectors passed to your system should be by constant reference.
• You may use the standard algorithm find and sort if needed.
• You may not use any statistical package to perform this computation as manipulating vectors is the main goal of this project.
• You must use vectors for ALL multidimensional data storage in this program.
• Your code must compile on cslab103 with the following compiler flags:

      -g -O2 -Wall -Wextra -Wpedantic -Werror --std=c++20   -Wnon-virtual-dtor -Wold-style-cast -Wunused-parameter -Wuninitialized  -Winit-self  -Wshadow  -Wparentheses -Wdangling-else 

• I was surprised that occasionally the null hypothesis will be rejected. Dr. Hoggard just laughed and said that was to be expected.

Required Files

Submission