What Distribution Does The F Distribution Approach As The Sample Size Increases?

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Suppose we have two samples with n1 and n2 observations, the ratio F = s12 / s22 where s12 and s22 are the sample variances, is distributed according to an F-distribution with v1 = n1-i numerator degrees of freedom, and v2 = n2-1 denominator degrees of freedom.

For instance, if F follows an F-distribution and the degrees of freedom for the numerator are 4 and the degrees of freedom for the denominator are 10, then F ~ F4,ten. For each combination of these degrees of freedom there is a different F-distribution. The F-distribution is most spread out when the degrees of liberty are small. As the degrees of freedom increase, the F-distribution is less dispersed.

Properties

The F-distribution has the following backdrop:

The mean of the distribution is equal to v1 / ( v2 - 2 ). The variance is equal to [ v22 * ( v1 + 2 ) ] / [ v1 * ( v2 - 2 ) * ( v2 - 4 ) ]

The F-distribution is skewed to the right, and the F-values tin be just positive. The curve reaches a top not far to the correct of 0, and then gradually approaches the horizontal axis. The F-distribution approaches, just never quite touches the horizontal axis.

Uses

The principal employ of F-distribution is to test whether 2 independent samples take been drawn for the normal populations with the same variance, or if two independent estimates of the population variance are homogeneous or non, since it is often desirable to compare two variances rather than two averages. For example, higher administrators would prefer two college professors grading exams to have the aforementioned variation in their grading. For this, the F-test can be used, and after examining the p-value, inference can be drawn on the variation.

Assumptions

In order to perform F-test of 2 variances, it is of import that the post-obit are truthful:

The populations from which the two samples are drawn are normally distributed.
The two populations are independent of each other.

If the 2 populations have equal variances, then s12 and s22 are close in value and F is close to 1. Only if the ii population variances are very different, s12 and s22 tend to be very different, besides.

Choosing s12 equally the larger sample variance causes the ratio to be greater than 1. If s12 and s22 are far apart, then F is a large number. Therefore, if F is shut to 1, the bear witness favours the zero hypothesis (the 2 population variances are equal). Just if F is much larger than i, then the evidence is confronting the null hypothesis, and nosotros can infer that possibly the population variances differ to a large extent.

Anova and F

In the technique known as Assay of Variance (ANOVA) which plays a very important role in Pattern of Experiments, the variance ratio examination is practical to test the significance of dissimilar components of variation against error variation.

For example, a new drug for treating Osteoporosis could demand to be field tested. Since severity of this disease is more often than not a part of historic period, the new drug could be administered randomly to n patients in each historic period group. Put differently, this would be an experiment in yard age groups and due north different dosage levels of the drug allocated randomly to the patients. With figures provided from patients for each age group x dose combination, nosotros can utilize the variance ratio examination (F- test) to examination for difference between dose levels and if this variation can exist attributed to chance.

The other uses include testing the significance of the correlation ratio betwixt two random variables, and to test the linearity of regression.