SIMULATION OF THE SAMPLING DISTRIBUTION FOR VARIANCE:

The sampling distribution for variances can be associated with a Chi-square (χ²) distribution with (N-1) degrees of freedom, where N is the dimension of the sample. In the case of variances, the expression used as the statistics is:

(χ²) _i=(N-1)*(Sx)²/ σ²

Where the symbol I represent the statistic associated with each sample, N represents the dimension of the samples, Sx represents the standard deviation of each sample, and σ represents the standard deviation of the population.

We used the following program for the TI-84 to make a simulation of a sampling distribution for variances:

Prompt N,R,M,S

ClrList L1,L2

For(I,1,R)

randNorm(M,S,N)->L1

stdDev(L1)->X

(N-1)*X^2/S^2->L2(I)

Disp “I=”,I

END

We selected samples of dimension N=6, the parameters for the normal distribution were µ=100 and σ=10, and the number of repetitions was 500.

In the following graph, I compare the result of the simulation with the calculations using a Chi-Square Distribution with 5 and 6 degrees of freedom.

As we can see from the graph, the results of the simulation coincide with the calculations with 5 degrees of freedom, as expected.
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