Cooper City High School
Broward Community College

Statistics

Comparing the t-statistics with the z-statistics :

Normal Distribution (Gauss Distribution): This is a probability, symmetric distribution that can be represented by the mathematical function:

sometimes, it is better to work with the standard normal distribution that can be obtained using the transformation;

where this variable is called z-value. This transformation is equivalent to use the values μ=0, and σ=1. Using this transformation we get the standard normal distribution:

Student or t-Distribution: This is another probability, symmetric distribution that can be represented by the mathematical function:

as you can see, we have a new parameter (ν) usually called- degrees of freedom(d.f).

The gamma function Γ(ν) is related to the factorial function this way:

Γ(ν+1) = νΓ(ν)=ν!

It is possible to show that when the number of degrees of freedom increases  the t-distribution gets closer and closer to the standard normal distribution.

We want to compare the results from the experiment with the calculations using the z statistic when we assume that we know the standard deviation for the population σ, and the t-statistic when we use the standard deviation from each sample Sx. First, let's perform an experiment with samples obtained from a skewed distribution. We select samples of 30 individuals from a randBin(5,0.1) which is a totally skewed distribution with mean value µ=0.5, standard deviation σ=0.671, and skewness =1.192 (totally skewed to the right). We repeat the experiment 500 times. The sampling distribution for z and t were calculated using the mean value and the standard deviation from each repetition. The results are shown in the following table:

 Mean Value Standard Deviation Skewness z 0.028 1.005 0.262 t 0.099 1.081 -0.469

Let's compare the graph for the z-statistic sampling distribution with the graphs for the  normal and student distributions.

Let's compare the graphs for the t-statistic sampling distribution with the graphs for the  normal and student distributions.

As you can see, there is not a big difference when the dimension of the sample is not small. Let's repeat these experiments using smaller samples. In this case, we have to select samples from a symmetric population, in another case we know the sampling distribution will be skewed where neither the normal nor the t distributions work.

We will select samples of 5 individuals from a randNorm(0,1) which is a totally symmetric distribution with  mean value μ=0, standard deviation σ=1, and skewness γ1=0. We will repeat the experiment 500 times. The parameters for the sampling distributions for the z-statistic(z) and t-statistic(t) where:  z= (x - μ) / (σ/√n)       and  t= (x - μ) / (Sx/√n) are calculated using the values of x and Sx from the experiment. The results are shown in the following table:

 Mean Value Standard Deviation Skewness z -0.0931 0.984 0.251 t -0.134 1.383 -0.164

Let's compare the graph for the z-statistic sampling distribution with the graphs for the  normal and student distributions.

In this case, the Normal Distribution describes the experimental results much better thant the t-Distribution. Now, let's compare the graphs for the t-statistic sampling distribution with the graphs for the  normal and student distributions.

In this case, it is clear that the t-Distribution describes the experimental data much better than the Normal Distribution.

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