Question

T has a t distribution with b degrees of freedom. T² therefore has an F distribution with 1 and b degreees of freedom. Using the formula for t distribution and transformation theory, find the density function of F=T². You must start with the complete formula for the t distribution.

Be careful as this is not a one to one transformation. Verify that the derived solution is the formula for F distrubtion in the case that a=1.

This shows a relation between the critical points in t and F charts. Compare the 5% significace point for t sided t test with 20 degrees freedom and the 5% significance point for an F test. Explain the comparison.

Answer 1

Let s shows the sample standard deviation, shows the populaiton mean, shows the sample mean, is population standard deviation. and is sample size.

Chi sqaure distribution with (k-1) degree of freedom will be

Here we will use following relationships between the variables:

Relation between standard normal distribution Z and chi-sqaure distribution:

Relation between F distribution and chi-sqaure distribution:

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Relationship between t-square and F:

t-statistics with degree of freedom will be

Now squaring both sides gives:

Hence, proved.

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Critical value of t for df=20 and 0.05 significance is: 2.086

The critical value of F for df1=1 and df2 = 20 and 0.05 significance is: 4.351

Since so results verified.